Basic Xbox headsets are cheap pieces of equipment that usually stop working like that. I've gone through four, all of which have stopped working randomly. What I'm trying to say is that that is a very common problem. They are not covered under warranty and any cost of Xbox repairing them would exceed the cost of the headset itself.
The solution is to throw it out and but a new one, and fully expect that it might stop working.
If you want to, invest in a higher quality headset, that won't just stop working, and that will have a warranty, if it should break.
