Yes but you first need to find out what is causing it not to work. Usually it is a broken connection due to the wire being yanked. If so, you will need to take it apart carefully to find where the break is and solder it back together. If the wire itself is broken in the middle, you can either splice it back together or replace the wire. You can find all different thickness wires at Radio Shack.
All the time and energy is not worth it. They're flexible, therefore, flexing wire causes the metal to become stressed. Yes, you can repair it, but the stressed wire will eventually break in another spot.