This should work but there are no guarantees. Doing any repairs like this may void any warranty you may have. Soldering irons and other tools can be dangerous and should be used with safety in mind. I am not liable for any damage done.
Here's a fairly well-written 'ible on the wire splicing
Step 1: What You'll Need
-Soldering iron and solder
-Shrink tubing (Finally found some in my house)
-Sandpaper (grit doesnt matter too much, probably something higher than ~140 should be good)
Step 2: Strip the wire
Step 3: Slip on Shrink Tubing
Just slide it on before you twist any wires together.
Step 4: Sand the wires down.
btw my camera really doesn't like the close-ups
Step 5: Twist and Solder
Step 6: Cover Solderings with Electrical Tape
Simply cut a small piece of electrical tape and wrap it around each wire. Cut/Fold away the excess.