See if this helps you:
https://support.mozilla.com/en-US/kb/add-a-device-to-firefox-sync

Thanks for the tip but unfortunately it doesn't help. I have been through these steps a bunch of times, but because I can't login without resetting my key (and subsequently deleting all of my backup), I never see the "Manage Account" drop-down or the "add a device..." option.

After attempting with the key I had saved and I get the "Wrong Recovery Key" message, there is an option that says "I have lost my other device" which will give the option to change the key along with a friendly note that says if you change your key everything will be lost!?!!

My guess is that my key is missing or has a wrong character in it. Since I wiped my computer, I cannot see it again. If the key is gone, does that mean recovery of the backup is impossible?

Try typing the Recovery Key without the 5 hyphens.