Hafting Nephrite Drill Bits

Is this the right way to haft one of these nephrite drill bits?  It works and it seems to suit the design of the originals, but I've never actually seen one hafted into the drill shaft to say for sure.  I make a step cut in the wood shaft and tie the drill in place with a bit of glue in between.  But its just a guess.

This design seems to work fine, but I don't know if it is ethnographically correct.  

This is one of the artifacts that I use as a reference for these Labrador Inuit style drill bits.

The drill bit itself is round, but they are set with a square or rectangular base.  

They aren't thinned at all , so I don't think they would have been hafted into a split shaft socket like a chipped stone drill bit (top).

Disclosure: I normally use these with school kids, so I don't usually  haft them with traditional glues or lashings.  I use artificial sinew and epoxy for the added durability.  So I can't really say how they would hold up if I used something like real sinew and hide glue or baleen and seal blood glue.

Photo Credits: Tim Rast