You can usually unwind a turn or two of the fine wire and reconnect it to the lug. Make sure to leave a small loop in the wire before it gets to the lug to take the stress off the fine wire during normal operation.

That coil is pretty grungy. I would replace it in this case. As others have mentioned, the part hanging out the other end looks like the coil stop that has come way from its bracket.

These flippers have seen better days and look like they have never seen any love.

I would be doing a full rebuild - coils, coil sleeves, coil stops, new plungers and links at the very least.