How to get locked Mac Mini to connect to a new keyboard

I told my Mac Mini to forget its connection with my Bluetooth keyboard so I could use the keyboard with another computer. Then my Mac Mini went to sleep. Now I am logged out and there is no connected keyboard to enter the password to log in. I can't get to the Bluetooth menu because I would have to log in first, and I can't log in without a keyboard.

Is there any way to solve this without borrowing a wired keyboard?