Why not try an Easy Out screw extractor? It worked for me on an exhaust manifold bolt with poor access.
With the good access you have it should be straightforward.
You got lucky. Ease outs are good for one thing. Breaking off and making the problem worse. As they screw in, they expand the broken fastener and usually lock it down, causing the ease out to break. You talk about a hell of a time when that happens.