It turns out I had the pickup ground wire connected to the wrong part of the pickup. I connected it to the brass plate, and somehow, it still picked up sound, just not as well as it should have. The ground is there to prevent electrons from taking alternate paths, but I guess some of them still made it to the right place??
Anyway, this tiny bare wire is the start of the coil, where the ground wire should be connected. (The live wire is connected to the end of the coil.) I got it on, and I *think* it sounds right after that.