RFID coil - ends broke, can I resolder?
If you unwind one turn of the coil, you will detune the antenna.  At 125 kHz, the coil should have many turns and maybe this won't throw it off so far that it doesn't work.  But if you have two broken ends, you would be tempted to remove two turns.  That would be even worse.  You might be able to add some capacitance to bring it back in tune.
Reattaching the wires by, in effect, splicing in some more wire, will work, especially at this low frequency.  There are some mechanical issues with those #26 wires; they will seem huge and will want to pull out the tiny wires, but electrically, it's sound.  It's the approach I would take, especially if I really just wanted to see if that really was the problem in the first place.
Some types of insulation will burn off so you can solder, but many times it won't, and then you have to scrape the wire.  That's a challenge in itself.  I would not bother trying to apply more insulation unless I didn't think I could prevent the wires from touching.