I get broken vertical pipes quite often and found at ace hardware a small wrench (shape of a T), that at the bottom has a conical shape with sharp edges. You push the tool down into the broken part, and it "scores"/cuts a little bit into the broken part of the pipe and you just twist it out. Works like a charm. If you have this happen often, you can add two small right angle pieces before the vertical pipe (means getting shorter vertical pipes). The two right angle pieces are left a little loose...
