You need a pin chuck with a top quality modellers drill. I did this early on. First job was to fix a washer over the drill site with double sided tape to stop unintended damage.
Then slowly start the hole, until the drill finds its own way. It took me several evenings work to drill down the centre until sawdust started to emerge, when I knew I'd got to the bottom.
Then I inserted a triangular modellers needle file into the hole and it just wound out. Simple as that.
A pin chuck is a finger drill which you rotate between thumb and forefinger.
PS - those screw fittings are almost monkey metal - not hard, easy to drill and easy to snap off..........
