There is an article in the September/October Muzzleloader entitled "Fixing Nature's Flaws".
The section about making wooden plugs will work very well for you, providing you have access
to a 1/4 inch leather hole punch. Chuck it up in your drill press and use it to cut out the wood around the broken drill, and to cut a plug from a matching piece of wood.
I think you'll find it works surprisingly well.