There are two types of E.D.M. machines. One is called a wire feed E.D.M. and is generally used to manufacture stamping dies. The other type is called a carbon E.D.M. and is used in the manufacture of forming dies and molds. If you can contact a shop that makes plastic injection molds they will probably have a carbon E.D.M. machine which is the machine required to burn out unwanted hard material broken off in holes. You might also try a small diamond burr powered by a flex shaft. A company called Harbor Freight sells these items and can be accessed from the internet.
Since you need to save the pinion gear on the other end, you might also consider cutting the damaged part of the pinion shaft off past the damaged portion and turning a shoulder on that end. Then make a new pinion end with a hole in the other end to accept the turned shoulder. The new part will then need to be hardened and tempered to the correct hardness and then pressed over the turned shoulder with a staking set. A small amount of Loctite will prevent the joint from comming loose but if you should ever have to seperate the joint for a future repair, just apply heat to the area and the Loctite will come loose. The final turning and polishing can be done after the unit is assembled and the Loctite has cured (@24 hours).