Well, I got it working, but not by any real troubleshooting: I had to change hosts and reinstall, then move my backup over to the new server.
I think iPage put me on a machine that was very bogged down. WP-Login.php was apparently crashing the PHP handler. So, I’m at BlueHost now, and it’s running better than it ever has before.
I’d recommend not using iPage for this. It can’t seem to handle it.