i would procure the wire diameters used in the transformer, count off the windings while i un-wind it... then re-wind it with new wire.
honestly the transformer probably only has 100-ish turns on each of the coils... with one of them making a stop part way to make a tapped coil... so reproducing this transformer would be straight forward. the hard part would be removing the iron core from the wire bobbin without breaking it... but something that could be done with solvent or heat.
protip, I use old and/or physically damaged pinball coils... the part numbers most of the time are the wire gauge and number of turns.
for example:
26-1200 is a coil made of 1200 turns of 26 gauge wire.
23-800 is a coil made of of 800 turns of 23 gauge wire.
so on and so forth. this will get you a "roll" of known wire gauge.
but sometimes you need some tiny tiny wire, like 34awg or 36awg... you can often buy small quantities on amazon. just search up "##AWG magnet wire"
peel off a section of transformer wire (counting the turns as you go) then measure the diameter and look up the diameter on the "american wire gauge" wiki and and find the closest gauge. (for examples above 26ga is 0.0159in, 23ga is 0.0226in).
once you know how it's rolled on there, just re-apply new wire in the same fashion, and throw some Kapton tape over it for insulation.