Pair a bluetooth mouse with the keyboard only
Found a solution.


CMD+SPACE to open spotlight search.
Enter "mouse" and select the "Mouse" system preferences panel (probably the first item that comes up).
CTRL+F7 (or CTRL+Fn+F7) to change the way the tabs move focus.
TAB until "Set up Bluetooth device" is in focus. 
SPACE to press the button.
Turn the mouse off and back on to put it pairing mode.
The mouse should come up in the list of devices. TAB to focus it, then space to pair. 


Normally, by the time you get to step 2, OS X will automatically attempt to locate your mouse. In this case, the computer was already paired with some other mouse (which I can't find) so it didn't do that.