English spelling does not have a one-to-one relationship with English pronunciation, so it shouldn't be too much of a surprise that "Nike" does not rhyme with "bike" and "strike" (except for when it does—apparently, there are some speakers who don't use the "official", disyllabic pronunciation for the brand).

"Nike" is from a Greek word-form, unlike "bike" and "strike"

To answer the question about why: "bike" and "strike" are spelled with the "silent e" that in present-day English is used to indicate a "long vowel" pronunciation. ("Bike" is an oddly formed shortening of "bicycle", coined fairly recently; "strike" comes from an Old English verb that had long /iː/.)

The "e" in Nike serves a different function: as Kate Bunting mentioned, this word is derived from Greek νίκη/Νίκη, meaning "victory/Victory (capitalized when used to refer to the goddess personifying the idea of victory)". The letter "e" in English "Nike" is a transliteration of the Greek letter η (eta).

The "traditional" English pronunciation of Latin and Greek

The Great Vowel Shift established a "traditional English pronunciation of Latin" (also used for Greek words) that continued to be used as a system for some time

There exists a tradition of pronouncing Latin words in English with certain characteristically English vowel qualities, and a tradition of pronouncing Greek words as if they were Latin words. Janus Bahs Jacquet referenced the Great Vowel Shift in a comment; this tradition did originate from this sound change, but my understanding is that not all words pronounced according to this tradition were actually taken into English before the Great Vowel Shift (which according to Wikipedia is thought to have occured approximately between 1400 and 1600).

It became an established convention for pronouncing all Latin and Greek words used in English, including words that were borrowed at some point after the Great Vowel Shift (although alternative systems have also existed alongside this system for a long time).

In the traditional English pronunciation of Latin, word-final "-e" is pronounced like the "-y" in "happy"

Word-final "e" in the traditional English pronunciation of Latin and Greek is typically pronounced with the same vowel as the word "happy"; sometimes represented in the IPA as /i/. I'm not totally sure of the history of this pronunciation (in traditional British "Received Pronunciation", this is not quite the same vowel as "long e"; it's instead considered to be the same vowel as "short i"). But it applies to Greek names ending in "e" like Nike, Calliope etc. the same as it applies to Latin words like simile, sine or carpe.

(There actually has been a tendency for word-final "e" to be reinterpreted as "silent e" in a number of words or phrases from Latin, such as "vice versa", "bona fide" and "rationale". However, as far as I know, this kind of reinterpretation has not yet come to be considered standard for the name "Nike".)

Alternative pronunciations exist for some words from Latin and Greek, but not for Nike as far as I know

The "traditional English pronunciation of Latin" (and Greek) is also why the i in "Nike" is pronounced as an English "long i" (IPA /aɪ/). The i's in via and viva are sometimes pronounced the same way. A competing "restored" or Continental-European-style pronunciation uses the English "long e"/"ee" sound (IPA /iː/). The restored pronunciation and English-style pronunciation have competed for a while; I forget exactly when the English-style pronunciation stopped being used in Latin pedagody in England. (It seems to have occured before 1920: John Sargeaunt's description published that year of The Pronunciation of English Words derived from the Latin says "This pronunciation is now out of fashion".)

But I have never heard of /niːki(ː)/, /niːkeɪ/ or /niːkɛ/ being a common pronuncation in present-day English. The pronunciation of a vowel in a word from Greek or Latin according to one pronunciation system or the other is often inconsistent and hard to explain; e.g. restored /ɑː/ is almost always used nowadays in the word "drama", but fairly rarely used in the word "data" (a word that also shows variation, apparently especially in American English, between a pronunciation with English "long a"—which is what would be regular according to the "traditional" system—and one with English "short a").

The only alternative that seems to exist is the spelling-pronunciation /naɪk/ mentioned in the first sentence and in the comments above.

(The name "Irene", as Kate Bunting mentioned, also comes from a Greek form ending in η. The pronunciation ending in /n/ might be a spelling pronunciation; I think I have also read an alternative explanation of it as possibly being influenced by the pronunciation of the French form Irène. Final vowels in Latin/Greek words correspond to "mute e" a bit more often in French than in present-day English; another similar example is the word systole, from Ancient Greek συστολή, where the final "e" is mute in French but not in standard present-day English.)

History of the transliteration "Nike": my amateur findings

I'm confident that "Nike" is pronounced /ˈnaɪki/ in accordance with the "traditional English pronunciation of Latin", but as I mentioned, I am less certain about when this pronunciation became established, since the earliest use of "Nike" in English that the Oxford English Dictionary (OED) records is from 1846:

T. S. Carr Man[ual of] Classical Mythol[ogy] xvi. 144 Nike..or ‘Victory’, was the daughter of Pallas & Styx, and was greatly honoured by the Greeks... Sometimes she is repersented as borne in the hand of Minerva.

Actually, looking through Google Books there seem to be at least a few somewhat older examples:

• 1842, The Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge:

  ..and that dedicated to Nike Apteros, or Wingless Victory.

And there is a slightly older example of "NIKE" being used as the transliteration of the word in reference to the concept of victory, rather than the goddess:

• 1827-8, The Botanic Garden:

  A Greek etymology has been given to it [the name Veronica] from PHERO, to bear ; and NIKE, victory

The 19th century may seem a bit late for the first recorded occurences in English of a name that dates back to antiquity. My guess is that it may have been more common in the past to simply spell the word/name in the Greek alphabet, which used to be more commonly used in scholarly works in English than it is presently. And rather than giving a transliteration of the Greek, a translation as "victory/Victory" or a Latin translation "victoria/Victoria" might be given alongside (or simply instead of) the Greek-alphabet spelling.

Some examples of "Νίκη/ΝΙΚΗ" being used without transliteration in English texts from before the 1840s:

• 1813, The Monthly Magazine, "M. Fauvel's Letters from Athens":

  On the car is written ΧΡΥΣΟ; on the victorious Genius ΝΙΚΗ, and on the third one ΠΛΟΤΟΣ.

• 1819: Pantologia: A New Cabinet Cyclopaedia...:

  VICTORIA, one of the deities of the Romans, called by the Greeks νικη, supposed to be the daughter of Titian and Styx.

• 1823: "Universal Technological Dictionary", George Crabb:

  ...this æra is marked on coins with the inscription ΕΤΟΥΣ ΝΙΚΗ, i. e. the year of victory.

"Nice": an alternative transliteration that I don't think has been used much

A transliteration more along the lines of conventional Classical Latin adaptation of Greek loanwords into Latin would use C instead of K, resulting in "Nice" (compare "cinematic" and "kinematic" for a similar example of variation that is actually lexicalized). The regular pronunciation of this would be /ˈnaɪsi/, although I guess it's impossible to know for sure how people actually intended for it to be pronounced (some people apparently pronounce "encephalitis" with a /k/ sound, despite the spelling).

It's been hard for me to find evidence of "Nice" being used to refer to the Greek goddess of victory, because it has certain homographs that are more common: the adjective "nice", the French place-name "Nice"—which interestingly enough actually is derived from Greek νίκη—and the place name "Nice" that was used in the past to refer to the ancient city of Nic(a)ea in Anatolia, location of the Christian ecumenical council called the "First Council of Nicaea" or "First Council of Nice".

But "Nice" is used in at least one source as a transliteration of the name of this goddess:

• 1826: A Classical Dictionary: Containing a Copious Account of All the Proper Names Mentioned in Antient Authors..., by John Lemprière:

  VICTORIA, one of the deities of the Romans, called by the Greeks Nice, supposed to be the daughter of the giant Pallas, or of Titian and Styx.

As you can see, it appears to be a revised version of the 1819 Pantalogia entry that used "νικη".

The transliteration "Nice" for the name of a mythological human woman, and for the non-personified concept of victory, seems to occur in Bell's New Pantheon (1790) and in A Dictionary of Polite Literature, Or, Fabulous History of the Heathen Gods (1804):

NICE. Daughter of Thespius.
NICE MARATHONI. A Grecian anniversary observed by Athenians upon the 6th of Boedromion, in memory of that famous victory which Miltiades obtained over the Persians at Marathon.

The related name "Berenice" is often spelled with "c", as is the derived name "Bernice" and the probably-derived name "Veronica".

The transliteration "Nike" is also used in German, and has been for some time

As DavePhD pointed out in a comment, it is possible to find the transliteration "Nike" in use in German texts from before 1846:

This 1814 German Mythology and Religion dictionary has a big entry for "Nike" Wörterbuch der altklassischen Mythologie und Religion (etc.) – DavePhD

Another example of "Nike" used in German can be seen in the following text from 1841: Nike in Hellenischen Vasenbildern; eine archaeologische Untersuchung, by Georg Rathgeber.

This makes me wonder if the use of the spelling "Nike" in German scholarly works might have had some influence on the use of "k" rather than "c" in this name in present-day English. It seems at least possible to me, but I don't know how plausible it is.