I had initially tried a more complicated motor mounting method in order to accommodate the length of the wooden shaft, but then opted to glue the stepper motor directly to the cardboard disc, and instead just trim the wooden shaft down. Had I committed to doing this from the start, I would have trimmed the shaft down first before attaching the mirror to it.

Drilling a small hole into a wooden shaft would have been much easier to do if it didn't have a breakable mirror glued to it.