Note that I've never tried this, but this fellow lists a couple of DOS IR drivers which you might be able to use and/or modify.
http://www.mbudde.de/mp3player/mp3_download_eng.htm
You'll have to scroll down a bit to find the drivers.
If you're willing to try Linux instead of DOS, the whole thing should become easier, since everything has pretty much been done for you:
http://www.hpl.hp.com/personal/Jean_Tourrilhes/IrDA/
Another site a quick googling turned up:
**broken link removed**
Looks to have quite a bit of discussion, including circuits and info on Window/DOS software for IR remote control.
Again, I haven't tried any of these things, but it does look like fun. And I too have a stack of old machines in the corner which every so often get turned into appliances (usually routers or firewalls) and then retired again when I get sick of the roar of old fans.
Hm. The amplifier question I'm not sure on. At some point you're going to hit the limit of what your IR LED can pump out. Maybe use a few IR LEDS on the transmitter? Repeaters are also possible (at least, they're on the market) but I have no idea how tough it might be to make one.
Anyway, just some thoughts. Use or discard as appropriate.
Torben