I'd recommend using butt splices, electrical tape and heat shrink wrap.
You'll need the following:
Butt splices and crimpers
Electrical tape
Heat shrink wrap big enough to go around the newly repaired cord, which you then heat lightly with a big lighter (heat gun is preferable, but the lighter works fine, just don't torch it.)
All of this can be purchased at a Home Depot or Lowes in the electrical section.