When troubleshooting it's best to start from the easiest things and work your way up. Have you tried a new cable and charger on a different outlet? If that doesn't help, have you tried to clean out the charging port? They can get lots of lint, dust, and dirt in them over time and it can cause charging issues. The best way to clean a charging port is using thin tweezers to get out any gunk.

Like @vojtech59738 said any good high heat source will work fine, however a tool like a hot plate is the "proper" way to do it. You can get the "China special" off Amazon or one from iFixIt. As far as the exact way to do it on this exact phone this tear down should help: https://www.youtube.com/watch?v=Ms8UZ46-2l0 https://www.amazon.com/Soiiw-Microcomputer-Soldering-Preheating-200X200mm/dp/B082H12PPT/ CPB Heating Pad for Screen and Battery Replacement

Ok within about 5 minutes of asking this question I found a solution to my issue. On earlier models changing the RAM config bypasses the EFI lock on the next boot, which allows you to do a PRAM reset removing the EFI password. For others in the same situation here's what you need to do: 1) Remove one of your two RAM modules, if you only have one remove it then turn on the computer and after the "no/bad RAM" beep code turn the computer off. 2) turn on the computer and after the chime hold down command, option, P, and R at the same time, wait for the screen to go black. ...and that's it. It's really simple and it was easy to find online, I just managed to miss it on my first search.

I've not fixed one of these before so I'm unsure exactly what the full damage could be. If it's all caused by just the broken plastic you could probably just buy a broken controller off ebay and swap it, but if the damage is further you'll need help from someone who knows these devices better.

What happens when you press the power button? Does it do absloutly nothing or does it try to turn on and fail?

I am late to your question but I found a YT video of someone tearing it down: https://www.youtube.com/watch?v=AO8lKvBR...

It turns out it was router configuration issues; I followed the official Nintendo guides here and changed my 2.4GHz access point settings to match what was recommended. Not certain exactly what fixed it out of all the changes though.