Dave Hi – A few days ago I was browsing through F.W. Britten’s Horological Hints and Helps when I noticed something which might help. If you can get at both sides of the screws you need to extract (even if they are blind holes). You could use an oversized screw extractor.
What is a screw extractor? It looks like a ‘Gee’ clamp. One end of the ‘Gee’ has thrust bearing on it. The other has a normal screw thread with a point which is sharpened with three triangular facets which are hardened. To use it you screw it down onto the broken screw. You then tighten it by holding the threaded part still and moving the ‘Gee’ frame and tighten up the lock nut. You then turn the frame the other way. If the point has dug well in to the broken screw you may start it moving.
I have never used one. It is cheaper than a ‘sparky’ and if old Britten who was a ‘downy old bird’ describes it. It probably works.
Finaly remember what we think are special screws very often are ‘normal threads’ used out of context. I bought a French made kitchen tool. To clean it you had to unscrew something. The Hungarians did the usual, lost the screw. There was then ‘wailing and gnashing of teeth’. Down to the workroom, sure enough it was ‘0’ B.A. and not 6mm. They had used an obsure commercial thread rather than design their own.
P.S. if you can get at both ends of the broken screw, then you use two three faceted points and leave out the thrust bearing
Good luck
Dick.