Yikes... that's a real PITA

I had this happen but it snapped near the head rather than on the shank.

As demacoz said you might be in a little strife with the swarf getting through the engine, I think all you can do is keep compressed air on it as you drill (head removed).

Personally I would remove the head and mask the entire deck (masking tape or contact adhesive sheet [degrease the block deck with some spray spirits or 'wax & degreaser' Diggers brand to get the oil off so the tape or contact sheet can stick properly]). In the meantime get some WD40 or the like in and around the bolt and give it a few very light taps taps with a hammer to break and binding that has rusted in below. Keep applying the WD40 to make sure it really soaks in then begin your work of drilling.

You COULD weld another bolt head or lateral piece of steel to the bolt remains and use that to turn it out but it depends on what you have available.