Previously, I would get the "Connection to server failed" occasionally, but it was never a showstopper, like the condition I described above.
I tried stopping the service on the "Configuration" side, then re-starting with no success.
I then deleted the client, then re-installed. It worked after that. It showed the previously downloaded movies that were in the "Completed" folder and still had the torrents in the "Torrents" folder. I stopped and deleted them. All is now good again.
I'm seriously thinking of getting a UPS just for this NAS.