I once had a similar problem: rusted 100-year-old screw broken off in the action. I snapped a hack-saw blade in two and used one of the pointy ends to scratch out a slot. Patience is definitely an asset! After that I let it soak in penetrating oil for a couple of days. It came out relatively easily after all that.Soldering iron and a very thiny file to open up the screw slot. Lots of patience required.