Neil,
I was a machinist and machine shop owner for many years.
Like most problems, there are many ways to approach this. The stem of the extractor is long and prevents you from getting a chuck or the spindle of a mill/drillpress in there.
One approach to consider is to make a drill extension out of a piece of larger diameter round barstock (drillrod is o.d. ground and runs pretty true) like 3/4" or 1" and drill the end for an appropriate diameter for a small piece of carbide blank. I'm guessing but those pins are probably about .070-.080" dia? If so, maybe a 1/16" dia carbide blank would work out for a spade drill. A carbide spade drill is easily hand ground and will cut pretty hard stuff, including broken taps. But you can try a short high speed drill too. The large diameter will reduce the wandering of the drill.
Another way is to Elox (EDM hole drill) the part out. Most cities have some sort of Elox service provided to the auto repair industry to remove broken bolts.
For the replacement, I'd drill all the way thru, put the pin and extractor in place with the pin protruding, TIG it, dress it down.
