I'd try get a decent socket set screwdriver bit & the brace or "T" bar set up from the socket set , then grind a perfect screwdriver slot in the unbroken end of the tap then ,
Gently heat the whole item to boiling point , slip it in a vice that you can work at from above whilst using the socket set screw driver set up ,give it a squirt of oil and use the screw driver and a socket set to gently but with firm pressure from above at the vertical try and unscrew the tap . You may find that the different coefficients of linear expansion of the block and the tap are enough to make a tiny workable clearance to start movement .
If there is no movement heat the whole block to a nice cherry red to let it cool with out doing anything to it to anneal the metals . Now try and unscrew it as above then using a made up mild steel drill guide clamped to item dot punch the tap once only , now drill it out using the drill guide jig & a cobalt drill then re -temper or case harden the block AFTER you have completed the tapping .
You could also consider laying a lump of MS over the protruding tap end whilst it is on a solid surface and giving ONE ( only one ) fair smack on the M .S. with a one pound hammer ..it may move the tap back a smidgeon ..just enough to enable you to do the unscrewing as indicated above . I'd do it before I cut the screw slot to reduce the chances of the tap shearing at the end around the slot .
Thinking a bit laterally about some of the things I've done to remove broken /sheared cylinder head bolts etc . ...
Having cut the perfect screw slot in the tap end with a Dremmel type grinding blade you could use the drill press chuck to hold the screw driver bit at perfect right angles to the axis and use the operating lever of the press to apply pressure to keep the bit in the slot whilst turning the chuck gently but firmly by hand or very carefully with a pipe wrench etc. on the chuck part that is not moved when you use the chuck key .
The secret of it all is a continuous gently downward force applied to a well fitting quality Diamond coated cobalt or titanium screw driver tip that fits perfectly in the cut slot , turned with a constant smooth rotational pressure/ movement .