I've repaired a similar problem on other electronics. I picked up some
standard ribbon cable at the local electronics store. The stuff that looks
like a bunch of regular insulated wires all stuck together in a row. I
didn't actually use the ribbon, but stripped off individual wires. The only
reason for not using regular wire was that I couldn't find any in the really
small gage like the ribbon cable has.

I tracked the attach points of the broken connections and sometimes found it
easier to attach the new wires to locations further away from the cable
attach point and routing the wires around things. In order to solder the
very fine points I took a short piece of 12 gage solid copper wire from
standard house wiring and clipped the end at an angle. I tied the copper
wire to the soldering iron with some more copper wire to create a very fine
point soldering iron.