Help me get at the Internet, the conclusion
Internet is working now. Upgrading the router's firmware did the trick. We'd tried this before, but it had failed — turns out Firefox is better than IE at uploading files to routers, or something.
Many thanks to everyone, especially Brad, Gavin, and the guy living across from me who has the router in his room. Now I'm going to go play KoL for the next three and a half hours.