Ok....looking at the pictures, and reading your description of what happened, it looks like the solder points have become detached from the board. This is going to be really difficult to repair.
You have to find some way to trace the circuit path from the button contacts to the next available inline solder point, and then find a way to bridge the break. You can attempt this in one of two ways.
1) solder a VERY fine wire to one of the contacts on the button, then solder the other end to the next solder point along the circuit path....then repeat for the other contact.
2) some years ago, I owned a bottle of electrically conductive paint (called liquid solder I think)
You paint it onto the board over the brown areas that are present, then while it is still wet, replace the button and extend the paint track so that it reaches the button contacts.
In theory this should bridge the gap from where the (now missing) solder point met the circuit board to the button contacts......
Option 2 would be the easier option, but unless you can secure the button so that the contacts don't move, it'll turn into a temporary fix as any movement is likely to cause the paint to crack and lose it's conductivity........
Check this link for solder paint...
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