I cleaned everything out of IE and still had the same problem with CCleaner. I then uninstalled CCleaner and rebooted and then installed an older version of CCleaner. That one seems to work.
The problem didn't start until after I renamed one of my users. I don't know if the stalling had anything to do with either IE or with renaming the user either.
The reason I thought it might have to do with renaming the user, is because the first time it stalled that way, there was a flash of a file (like a file pathway in Windows Explorer) with the original user name from before it was renamed. Then it remained stalled awhile longer and then it deleted as usual. If that original user name hadn't flashed up, I wouldn't have made any connection to that being a possibility, but in actuality I don't know what's causing it.
Whatever it's doing, it seems to be waiting for something before it can proceed.
So far, I haven't had that with the older version I installed. That one is 5.05.5176