Clearly, your cat considers that the cat food you provide isn't of sufficient quality ;-)
If you can disassemble the headset you can use a testmeter to check for continuity from the mic itself (or close) right through to the jack and hopefully narrow down the fault. The multi-stranded wire used in headphones is tricky to solder and it's easy to get a tiny solder bridge, or one strand touching where it shouldn't, or it could even be a faulty jack. Also, check there aren't any shorts between any of the T, R, R and S - in fact, try that first! Or maybe the cat chewed somewhere else you didn't notice.