Here is a getto method for drilling a centered hole in a shaft:
- use 2 wood boards to build an L shaped structure - use 2
identical cordless drills, in one hold the shaft and in the
other hold the drilling bit - use the L shaped wood structure as
a guide for the drills to keep them aligned, place them upside
down facing each other, so the drilling bit is close to the
shaft that needs to be drilled -turn on both drills at the same
time and slide the one that has the drilling bit towards the
other one gently, but apply presure so the drilling bit starts
to drill the hole, retract when hole is deep enough Now not all
drills allow this method, since some don't align properly on the
L shape guide, but you can adapt this idea to your situation.