home button how to solder the connection to the mainboard

There are a few ways to fix this, but I would guess the easiest way would be to solder a 30 gauge wire from the back side of the home button (goldish colored contact) to where the metal spring use to be to make the connection. I can't tell if the contact was lifted or not in the photos though. If it is lifted and soldering it is impossible, you will need a multimeter and attempt to find a dead short connection to another location you can solder to using the good board you have in the second photo. It might be tricky to find a solder point, but I would think somewhere on the board there will be a connection.

Good luck with your fix.