Computer not picking up audio from headset

Yes, it's me again, with the exact headset that was giving me problems before.

I gave up back then (about a year ago), but felt an urge to record something so I'm trying to get the dumb thing to work once again.

Not much point telling you guys about my headset, because it's apparently made by a company in my own country (I just found out today. FML).

Interestingly, I managed to get the thing to work for about a minute after jacking up the mic volume to 100 and the mic boost to +20.0dB. Recorded a typical mic test (you know, "testing 1, 2, 3"), after which the damn thing promptly died once again.

I see no damage to the wires or the headset, the headset worked that one time when I unplugged the headphone cable (i.e. only the mic cable was in), but as I mentioned, it's stopped working like totally. Meaning that Sound Recorder isn't picking up anything.

Speaking of Sound Recorder, unplugging the mic jack and plugging it back in elicits some response from it, but that's probably par for the course.

Well, here's my question: How can I figure out whether it's a problem with the headset (which I think is very likely), or a problem with the computer (specifically the Sound Recorder program)? And if it's a problem with the headset, what brands/models can you guys recommend? I'm looking for a reliable mic that has decent (or good, preferably) sound quality at relatively low cost.

p.s. I have no other mics in my house, so I can't test it out that way.

EDIT: It just occurred to me that I should let you guys know that I'm planning to use the mic to record song covers. Or rather, my pitiful attempts at them. Which is why I need a good mic. It also doesn't HAVE to be a headset, a standing mic will work just as well.