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.