Edit #2: I just re-read and saw the bolt head is snapped off.
Is the head or manifold aluminum? (where the bolt is)
In your case. (you probably tried this already) I'd soak it in Liquid Wrench, repeatedly, for as long as you can.
Use a breaker bar or pipe on the end of socket or wrench to get as much leverage as you can fit in there.
If you can fit a drift pin right on top of the stud, I'd hit that with a hammer a few good shots, along with the liquid wrench.
The best way to get a bolt like that out is heat.
Usually 02 & acetylene. It will take ten seconds; you have to get the bolt red. I've had it work even using a small atctylene (yellow tank) The real stubborn, rusted bolts have to be red hot and the come right out.
Edit: Bring to one of those outdoor car washes with the spray wand. Wash all the grease off the engine first.
I know how you hate to get messy.
Good luck, at least you have nice weather.