"Dripping solder" is not proper solder technique. For proper soldering you heat the metal and apply the solder to it. You do not heat the solder and drip it onto the metal -- especially an electrical connection. The proper fix would be to cut the wire in the middle of the bare wire. Slip two pieces of heat shrink tubing over the wires, twist them and solder them and then heat the tubing. Or if you don't like to or know how to solder properly, use a crimp type butt splice. If both ends of the wire are easily reachable, why not just replace the wire.
