I had exactly the same issue, that the Apple wireless keyboard became unpaired, and I couldn't get past the password-protected log-in screen, because I didn't have a wired keyboard laying around.
Discovered by trial and error that it was the wireless mouse (not an Apple brand) that was preventing the keyboard from pairing, even though it's not a bluetooth mouse...
Here's the steps I used...
1. Disconnected the USB mouse cord from the Mac Mini.
2. Turned off the Apple wireless keyboard from switch on the back.
3. Turn Mac Mini off, then back on.
4. Waited for it to boot up as far as the password-protected screen.
5. Turned on the Apple wireless keyboard, waited for it to stop blinking, indicating (hoping) that it paired up.
6. And voila... was able to then type in the password...
7. Then plugged the non-Apple wireless mouse sensor cord back into the Mac Mini's USB port
I also immediately went in to the MacMini's Sharing options and set that up for remote management in case something else happens down the road for other problems...
Thanks though to all who suggested solutions... they may come in handy in the future.
Hope this helps someone else...