The 'battery pack' you cut off isn't just a battery pack, it's also a controller that can produce partial string lighting effects. The lights are connected in two reverse paralleled groups for that reason.

To light all of them, you'll need to reverse the polarity quickly enough to give the visual effect of all LEDs staying on. A few hundred Hz is fast enough for most people, but kHz should be good enough for everybody.

If you don't want to build something from discretes, then a H-bridge motor controller, or maybe even a class D audio amplifier module could be pressed into service.

As you're wiring it to a power supply, you could use an AC power supply of either the right voltage, or one of higher voltage with a series resistor to drop the voltage. This will mean that the alternate strings of LEDs light only at 50 or 60 Hz, depending on your location. This is slow enough that almost everybody will notice a perceptible flicker, but as they're a decoration rather than 'lighting', maybe you'll get away with it. It's by far the simplest solution.