Touch ID works but Power Button does not.

Sadly, replacing the logic board maybe your only option!

Apple efforts to make access to one's data as limited as possible so the power button and logic board are bonded together so one can't use another power button as the TouchID sensor within the power button data link to the logic board will no longer handshake properly.

The physical switch is a simple blister switch but the flex cable from the button is quite fragile so it's possible it was damaged. In either case almost microscopic repair would be needed to either replace the switch or patch the flex cable. If either fail to fix things then it would be tome to replace the logic board.

This is the 2016 model TouchID/Power Button for reference yours will be similar

Maybe you can find someone with the deeper micro electronics skills to fix your button that is if that's the problem. I would dig a bit deeper, what does the diagnostics tel, you? Restart the system and press the D key to enter. Review this Apple T/N Use Apple Diagnostics to test your Mac let us know what if any errors popped.