Have you tried all this with it plugged into power? Battery might be dead.

Dell laptops have issues with the power regulation chip. What you need to do is pull the back cover off and unplug both the main battery and the CMOS battery. You need to be very very careful removing the CMOS battery. Do not pull the battery out by the wires. I find it’s best to use a plastic pry tool of some form, whether it be a spudger, old credit card or some other thin semi-semi blunt plastic tool. If you rock the two sides back and forth then push out the connector down the center, it will come out. The CMOS battery looks like this: https://guide-images.cdn.ifixit.com/igi/rWTpWLLvSdZ36TQU.large Once you have both batteries unplugged, hold the power button for 60 seconds. Yes, really, 60 seconds. Plug both batteries back in and plug in the charger, then attempt to turn the laptop on. If all went well, the CPU fan should spin over a few times before the machine finally boots back up. Put the cover back on and you’re done. Source: I’ve been working in IT Support/Management with Dell as our main vendor for the last 12 years.

It looks like what's broken is the retainer just above the power button itself, which in turn appears to be part of the palmrest assembly (compare picture below).  That's likely what you'll need to replace. https://www.parts-people.com/index.php?action=item&id=26064

Hi MCWERP, What is the exact system model? Does the system overheat? Is there any damage / liquid spillage on the system? When you press the power button, does the power led come on and stay on or does it go off after a few seconds?  When you plug in the adapter to the system, does the adapter stay firm or does it fall off easily? Is the adapter frayed? Please check if there is any damage on the machine. Check if all the ports on the machine are fine or misaligned. Sometimes if the pins in the USB ports are touching each other, they are known to cause many issues on the machine. Disconnect the adapter and battery and press and hold the power button for 30 seconds to release any static power. Then test the system with the adapter only / battery only. Press and hold the fn key + power button to turn on the system and it should initiate the diagnostics - check for any possible beep codes. Run the LCD bist - Turn off the system and press and hold the 'D' key + power button to turn on the system and it should run the LCD built in self test - see this link for more info - http://dell.to/1SkXNrG Enter the service tag and download the service manual and open the chassis - remove the HDD / memory modules and check if the system powers on and detects the absence of memory indicated by beeps. If nothing works, I would suspect a motherboard fault and it needs to be replaced. If there is no warranty, then you could contact our team to get a quote for a paid service call - http://dell.to/1vnT6CQ Also, please click my DELL-username and write me a private message with the service tag and your Name for case records. Let us know if you have any other queries.

Could you check your manual to find where the power button is?