Probably the best thing you can do if you really want to try to fix the corroded traces is to take a multi-meter and a thin marker.
After each trace that you check with the multi-meter for continuity in the circuit, use the magic marker over the one you've checked as good.
There are quite a bunch that run thru the board to the back side and it's easy to get lost if you do not mark the ones you've checked as "good".
At some point, you will find a trace that is disconnected or corroded right thru - that will be the one you will need to solder a jumper wire to replace
the broken trace from point A to point B on that line.
Concentrate on finding continuity in the traces in the area around where that battery was....use a magnifying lamp and take your time.
You will find the broken trace.