@Lord of the Rings:

Thank you so much for your help!

So, here's what I tried, and my results:

- I spend some time tinkering with Windows XP firewall, and the first thing that I noticed was that (contrary to what I had originally thought) it wasn't actually even on! I do have ZoneAlarm installed, so I turned that off temporarily to eliminate the firewall from the equation. I still noticed the same download speed problem, though.
- The magnet link worked just fine for me. LimeWire found 2 hosts almost immediately, and reached a maximum download speed of over 170 KB/s very quickly.

I then thought that it might be the actual items that I was downloading that were causing the problem, and this theory actually seems to hold water. I tried downloading some different items at random, and I was able to get the old behavior back - I connected to hosts relatively quickly, and reached download speeds above 10 KB/s.

Could it just have been the particular items that I was downloading? I found hundreds of results and potential hosts (many items having nearly 100 sources), but nearly all of them either stalled on "Connecting..." or started downloading, but maxed out at 0 KB/s and just trickled in.

Thanks again for your help - please let me know if you think it could have just been the particular items that I was downloading that were causing the problem.