1)If the broken off threaded bit is reasonably near the outer end of the hole (not recessed too far), you might try using a sharp scribe - stick it down the hole offset to the side, and with as little downward pressure as possible, and still not having the end of the scribe slip, try try turning the screw CCW.
2)There's another way that may work if you have the broken off knob and the remaining shaft is not bent. First put the broken knob shaft in the hole and ascertain where it "fits" against the broken off part (it will likely be broken off at a slight angle and you'll need a snug fit for the next steps) - mark the knob with a felt tip so you can place it back in the proper orientation in shaft hole later.
If the broken bit is recessed, wipe a small amount of Vaseline or grease on the shaft of the knob - the purpose of the Vaseline is to keep glue from migrating on the threads - if you're careful you don't need this step. Next put a tiny spot of super glue on the very end of the break on the knob . . . it must be a small amount as you don't want any glue migrating to the threads. Orient the knob with the index mark and place the shaft into the hole. It must fit properly and not be disturbed for several minutes. Attempt to unscrew the knob very gently, avoid any lateral force on the knob.
3)A machinist might try a left hand twist drill. The idea is that the left turning drill bit will very likely remove the broken piece. But if it does not, then a small "easy-out" is used to remove it. Warning - the drill bit must be small enough and stay centered on the broken screw so as not to damage the female thread in the hole.
Good luck!