The damaged windings are probably quite near the surface so one other thing you could try (still fiddly) would be to unwind the top few layers of winding until you reach the last point where it's broken. Then connect this bare end to the corresponding output wire. You'd end up with a few windings less than before but probably not enough to make a significant difference to the tone or output. Might be worth a try before you go for the rewind, not a lot to lose!