You have two options, neither is a very pretty solution.
One, cut the lead before the section of damaged wire (or close to the base if you aren't sure where the break is) and solder on a new lead, covering the exposed wire with heat shrink.
Two, open up the actual casing and solder a new cable in place. Unfortunately, this method actually breaks the case and you'll need to use electrical tape to get it back together again. Also this is probably not techincally legal, unless you are a certified electrician.