Start without power buttom

Sorry don't have any good answers which can solve all of your problems.

To start off if you are able to get the system running in its current state I would immediately backup your system so you don't loose all of your stuff and the current setup.

Instead of trying to gain direct access if you can get the system to power up since you state the Power/TouchID button is not working, if you have a second Mac system which you can connect to and have access to same iCloud account you should be able to treat the MacBook as an external drive to salvage your data.

So let's get into the needed repairs a bit...

Clearly you need a new battery at least, and if the Power/TouchID button is damaged replacing it will require a full system wipe to then connect a fresh Power/TouchID button only Apple can do that. But it does sound like the logic board still has issues which would require replacing or if you can find a talented Mac repair person who has the chops to dive deep into the circuitry they maybe able to repair the logic board instead.

You'll need to tally up the costs and decide if the outlay of funds is worth it! You might find a used system which is newer of if you can live with a 13" display the Neo is quite nice!