Apple Support worked me through this. Here are the steps as best I remember:
1) Turn off all running Apps.
2) Go to Settings>iCloud.
3) Sign Out of iCloud. Sign Out is at bottom of the column. I had to scroll down on my iPad Mini to reveal it.
4) You'll get a prompt, "Sign Out of iCloud." Click Sign Out.
5) You MIGHT get a prompt such as "Turn Off Notes." Click Delete.
6) You will get a prompt, "Delete Account." Click "Keep on My iPad." The machine will do a safe save (which means it will make sure all apps are properly closed, even if you can't see them running).
7) Enter your Apple ID Password. Click "Turn Off". Wait while the iPad saves.
8) Close the Settings app.
9) Power down the iPad.
10) Restart the iPad, then unlock the screen.
11) Go to Settings>iCloud. Sign in with your Apple ID on the top line (where it says "example@icloud.com") and password (where it says "Required") Tap "Sign in"
12) You'll get a prompt, "Merge with iCloud" Click "Merge". I got another prompt, "Find My iPad Enabled." Click "OK"
13) Settings>iCloud>Storage should show your available storage. Mine went up to 50GB. (It finally recognized the additional storage I bought.)
14) To run an immediate Backup, go down the iCloud screen's menu to "Backup." Tap it and you'll go to a screen where you can click on "Back Up Now." Or, you can just wait until an automatic backup runs when Backup is on, the machine is plugged in to a power source and the machine is locked.
NOTE: I did an immediate backup and it took MUCH more than the "estimated time" of 5 minutes, but this was the first successful backup I'd run in a long while.
If all of this doesn't work, you may have to Chat with Apple Support. Kevin McReynolds, the Apple Senior Adviser who helped me, said he did one additional step (not seen by me) from his desktop. But I believe the steps above will work without an Apple operator's intervention.