Wireless keyboard no longer connects

Since a recent update, my Apple model A1644 wireless keyboard no longer connects to my iMac, the one that has no connection ports. I have charged it, turned it off and on, rebooted the iMac, unplugged it and then rebooted again, and verified that the keyboard does, in fact, work by connecting to a MacBook Pro and logging in. My Magic Trackpad works with the iMac, but I cannot enter any data in any field with the keyboard, so I cannot let the system do its thing.

Any suggestions? The “Related Article” and several KBs were not helpful because most of them require connecting a cable to the Mac, which I cannot do.