I've heard that the heat can damage the diaphram, so it may be better to get a replacement part from the manufacturer. I would have quickly replaced my sennheiser cable if not for such an annoying roadblock.
You might was well give it a try. Either that or send the headphones in for repair.
Be warned, soldering headphone wire can be tricky because they use an odd type of cotton/wire for flexibility that doesn't solder very well. It can be done but don't expect it to be as easy as soldering a DIY amp.
My dog chewed some headphone cables and even though I was able to splice them back together, I was never totally happy with them afterwards and had to buy new ones. Nothing wrong with the sound mind you, I just couldn't get over that they weren't 100% anymore.