Wireless iMac keyboard will not let me type during sign in

Please excuse the basic tone of this but, as these are common causes, I have to ask:

• Did you find the switch that turns on the keyboard?
• Did you connect the cable to charge the keyboard?

https://manuals.info.apple.com/MANUALS/1000/MA1738/en_US/magic-keyboard-userguide.pdf

You will need to borrow/use any USB keyboard to get past the sign in. Then go to System Preferences > Bluetooth and pair your keyboard with the Mac. After that, it should be fine. By the way, it is handy to have an old wired keyboard stashed away for just such incidences.

👍