Yesterday I went to talk to a couple of techs at the help desk at our university. At the end of a very long session one of them asked if I had completely uninstalled and reinstalled Office. I replied that I had not because, after at least five online repairs, the problem was not resolved and because I didn’t want to get involved in a lot of reconfiguration issues for nothing.
When I got home I decided to try completely uninstalling and reinstalling Office365. To my great surprise, after I did this, the iCloud installation worked fine after I did a repair through “Programs and Features” in Control Panel. A secondary surprise was that Office stored all my settings and reused them after my reinstall — two unexpected pleasant surprises!