Did the chain actually break? Did the cable fray and break? Almost every failure I've come across was caused by the failure of the solder holding the brass sleeve to the cable. Nothing broken. In this case you can remove the cable and resolder the sleeve. That part of the cable where the sleeve lives is cleaner and there may still be some solder on the cable. Wire brush and flux the cable, locate the sleeve and heat the sleeve - solder will wick into the sleeve from each end.