OK, I got it working again, but I'm not sure how. I rebooted and went into "Safe Mode with Networking". It worked fine from there. Then after rebooting normally it seemed that all was well again.
One thing I noticed after going into safe mode and back to normal is that the remote desktop window showed up on my primary monitor (I have two monitors). When it wasn't working, the remote window would show up on my secondary monitor and would hang when I tried to move it to the primary.