All good suggestions. I am interested to hear how you replace this tuning pin. Like others here in over forty years of working with autoharps and hammered dulcimers I've never seen that happen to a pin.
"The dulcimer sings a new song."
It looks to me that the string hole is now more-or-less a slot. First, I'd try using a screw driver, but not really expecting that to work.
My second try would be to deepen the slot with a saw and files. This would require removing other strings and maybe some pins. I think that would require less skill than drilling for a screw extractor.
It also looks to me like the string hole in the broken pin might have been off-center. I would also be concerned about the variation between notes in the string windings.
If the pin holes do not go all the way through the wood, some may have bottomed out. The pins may not have been backed out enough before winding on the new strings.
There is enough of the pin sticvking out of the hole that you can unscrew it. I would use a pair of locking pliers, such as Vice-grip pliers, clamped onto the pin from above. Gently twist the pier handle counter-clockwise to unscrew the pin. Don't rock the pliers back and forth and there should be no damage to the hole. That is a very strange break. It will probably never hppen to you again.
Is that a Hammered Dulcimer? In some decades of messing about with dulcimers I've never seen a steel pin shear like that. Wooden pegs, yes, but not autoharp pins!
The tuning pins are threaded into the wood of course. There are "screw extractors" which you can purchase to aid in the removal. Basically, you'll have to use a tiny drill bit to drill down into the broken tuning pin, insert the the screw extractor (which IIRC has threads reverse to normal), then unscrew the pin using the extractor. Do an internet search on 'how to extract a broken screw".