Oh don't misunderstand me KMoffett, I don't want to endanger anyone, under no circumstance
The first option will always be getting a decent spare part or a new unit. But in some countries the spare is really hard to find, and the girl may not be happy using another cheap straightener, even if it does the same thing (you know how complicated the girls are).
So, If the OP is good with soldering and twin epoxy, he can resolder and then rebuild the plug jacket. Of course, checking the integrity of the finished job by bending and pulling the cord is a must!