[morzh]

Is the wire brittle? That could be.

It is not too bad rewinding these coils at all.

First, either make many photos, or hand draw a picture showing how the coils are, how they relate to each other, what length each coil has and what gauge wire is used.

Then not the terminals the wires go to AND !!! the direction of each winding.

Then unwind the coils, counting turns. Turns are not partial, they are always whole, so there will be a whole number of turns in each coil.
Write them down.

Then wind the coils with new wire, leaving several wraps around each terminal just in case you break the wire when soldering - you can unwrap a turn or two and solder again.
Then solder.

If one coil goes over another, solder the first coil before winding the second. In case your soldering damages the lead such that you have to rewind, at least it will be just one coil.

It will take you no more than 30 minutes for the biggest coil.