I visited the Apple store this morning, and one of the Geniuses was able to revive my iPhone. I don't know exactly what happened, and even if I'd asked I'm not sure he would have been able to tell me. The important thing is he was able to get it into restore mode and get it working, then restore my data and such from the cloud.
Since I hadn't backed up to iCloud in a while I lost some photos, but otherwise everything is back the way it was. The tech mentioned that if I use wi-fi at home (something I don't tend to do, as I'm not a big data user in general), the phone will back up automatically whenever I plug it in to charge, so that's going to be part of my regular routine now.