I think your last suggestion is what I'll go with. I unplugged all the networking equipment for a few minutes, so we'll see if that helps. I'm reasonably certain that there's 15 static IPs at most, but no, the record keeping here is not good. I was lucky just to find enough information to properly configure the router in the first place. I also have a full 24-port switch with no labels whatsoever, so I know that most of them go to one room, and a few of them go to these other few rooms, but I couldn't identify a specific one. I hate non-labeled cables.

Anyway, it's looking good now. Thanks so much for all your help guys. Matthew_k, you mentioned one place to find information, but if I wanted to learn more about networking (other than in the middle of a crisis with 40 people looking to me to help them), where would you recommend? Online is preferable, but I'm willing to get a book if it's really good.