I would just try doing it myself since it's really not hard. In fact since it looks like your dog chewed it in the middle you don't even need to buy new wire, you can just splice the two sides together. Just google "how to splice wires" and you'll find some good tutorials. Basically though, all you'll need is a wire stripper, a soldering iron, some solder, and some electrical tape. All you really have to do is peel back the the insulation on the cables and then on the individual wires until you expose the bare metal, then all you have to do is braid the matching sides together (or get them in contact some way) and keep them in contact. So actually you don't even need a soldering iron or solder, you could just use electrical tape (or any tape really) and have it working. The soldering iron and solder just make the connection really strong (stronger than the orginal wire if done correctly). I hope that made some sense.
TLDR = there's nothing special about headphone cables or fixing a broken one. As long as you can get the corresponding metal wires on each side to touch each other and stay in contact your headphones will work.