Repairing a wooden chair

We have an old style wooden chair whose headpiece (the crosspiece at the top that the narrow vertical slats fit into) has broken at one end. The stile–the vertical end piece–that is supposed to fit into a hole on the underside of the headpiece. None of the other pegs or holes are affected. I want to reattach the headpiece with wood glue, but the broken part (broken flush with the surface of the headpiece and the top of the stile) presents a problem. Is it feasible to drive a short, thin nail into either surface to reinforce the connection?
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I had the same issue with a chair at my parent’s house.
I drilled a hole in both the stile and the piece stuck in the headpiece, and used a short dowel with lots of wood glue to pin them together.

Agree with Beowulff. A nail will have little or no holding power. A dowel with glue will work best.

I just had another thought. If you can drill the broken dowel out of the top piece, and drill the same size hole at the proper angle into the vertical slat, you could possibly put in a new regular size dowel (with glue) into those two pieces. It would be difficult but not impossible to get the right angle into the slat.