How do I pair an already-paired Apple wireless keyboard in El Capitan?

I have seen several answers regarding previous versions of OS X that address this question using the Set Up Bluetooth Device option, but that seems to have been removed in El Capitan.

Every time I try to pair a keyboard that has already been paired with another machine, the pairing process fails on the Enter Passcode screen. The keyboard simply refuses to accept the keypresses.

I do not have access to the machine that they keyboard was previously paired with.