Can you restring an audio cassette tape which broke loose from one spool?

Yes. There used to be audio tape editing kits available to splice audio tapes together.

There also used to be high quality (very expensive) cassettes that were held together with small screws. You could unscrew the cassette and re-string the tape on it. I don't know if either of those things are still out there -- try a search for them.

If you can't find both of those things, you could take another cassette that has the tape good on both reels and unroll some tape out from one end. Cut the good tape that connects to the reel and splice your tape onto it. Use a pencil eraser or your finger to turn the cassette reel to roll your old tape onto the reel. It takes a long time, but it won't break the splice. Unroll the new cassette's other reel and splice the end of your tape onto it like you just did on the other reel. That will allow you to play the tape -- but you should immediately record it to a new tape (or computer or CD) so it doesn't get screwed up again.

Rod Stewart