Before using the paper drill I press the bit into a block of wax to clean & lubricate the bit.
I also use a soft book underneath so the drill bit does not dull itself hitting wood.
2 size holes are needed here. The small drill (1/8″?) is for shallow 1/2″ deep holes for switch posts, skewer holes, and hanging ribbons. The small also goes entirely through the cork for the eye hole (or nose) to allow the LED leads wiring.
The larger drill (1/4″) is to drill a shallow recess hole (1/4″ deep) for the LED to slide into. If you want your LED to be fully exposed you do not need the larger hole.
After drilling you may need to pick the core bits out of the hole with a stiff paper clip or dental pick. The synthetic corks they make today are fairly stubborn to remove excess.
Drill the holes you need for this design.