You can order left-hand drill bits from Northern Tool online. However, normal drill bits won't touch a tap and I don't think you'll find carbide left-hand bits anywhere.
The only time I've seen tap extractors work is when the tap broke because someone leaned sideways on the tap while threading the hole. If the tap breaks under pure torsion the extractors won't budge them.
Since this is a frame, the industrial way of removing the tap using a sinker EDM won't work. Frames are just too big. However, if you have a high-speed die grinder (not Dremel... an honest to God pneumatic die grinder) then you can use a carbide burr to grind out the broken tap. Common burrs have 1/4" shank diameter. You're best choice would be a conical solid carbide burr that doesn't exceed 1/4" diameter. You have to patiently advance the burr into the broken tap and worry it out of the hole.
Whatever you try, don't let anyone tell you to try and weld something to the broken off part. Taps are already brittle and the heat of welding will only make that worse.
EDIT: Darn... I posted 1 minute too late to be of any help!