Just got off the phone to my mate who will have a go on monday evening, hoping to weld a nut on and remove it like that, failing that he has access to quite a bit of machinery and next step would be carbide drilling, failing that it'll be cut it out, reweld it and retap.
Donnie, please, slowly, slowly catch a monkey. Go old school before welding, carbides etc or it'll end in tears.
Broken tap. What made it break? Crap tap? going hell for leather into existing hole? wrong tap wrench? Bottomed?
Knowing those things will tell you WHY it happened, and HOW to get it out.
EDIT: if space/ protrusion allows, always try a slot, to try a screwdriver, even if you have to use 5 x 8mm diameter slitters. (Crank/ grinding dust applies still!, see below)
Without knowing those things, the way we've always got them out is get all surplus material in the flutes out, even if you have to use a chisel in them (yes, a 0.5/ 1.00mm chisel), and a vacuum hose or compressed air as a poor second (compressed air only forces debris IN, hence never use compressed air on a lathe/ milling machine! etc.).
Your left over tap (if a good quality one-only) will give you it's own positions to make a 3 or 4 prong tool to insert INTO the tap flutes and unscrew. A bit like a security type tool. You don't need to make it go too deep (1mm), but you'll need as flat a surface (broken tap) as you can achieve. (grinding, obviously)[crank out!!!] grinding dust will destroy it (as will ANY weld splatter, a note for your mate re nut welding).
If a 4 flute tap, then a 1mm slitting disc will do the job handsome. If 3 flute, then again a 1mm slitter will achieve it but a bit more care required.
sorry been busy for a few days with work, yeah, not enough to weld to the tap so went to the carbides but only got so far so it's going to get cut out, rewelded and tapped.