Making a Davis kit dulcimer playable

Well, it seems a lady nearby has a Davis dulcimer she built from a kit many years ago, and wanted me to help finish it up for her. I came over with my tool bag full of tools, and thought it would be a 1 - 2 hour job at most.

Then I saw the tuning pegs! What was Bill Davis thinking when he designed those?

They look more like bottle stoppers than tuning pegs! They taper down from about 1/2" to about 1/4" in less than 3"! My tapering tool and reamer were about useless here. Got any ideas for us to try?