My advice is this: take it easy on the easy-out, they are not unbreakable!
Grind the broken easy-out flat on top to help keep the bit from wandering.
Get a short carbide die drill to drill with. You likely do not have one of those. A masonary drill can be made to work, but you will have to cut it off so it is very short and stiff (ie., rigid) for chucking close to the end. Masonary drills do not cut steel very well unless they are resharpened (on a green SiC wheel or diamond) to have some proper geometry, but mainly to create a style of split point to reduce the enormous pressure required to force the thing to make some dust.
I have also used worn out carbide burrs and ground them into a steel cutting spade drill type point (a basic thin wedge) and had a certain degree of success that way. The tool has to be retracted and cleaned very often because it has no flute to speak of.
You might also rig up some kind of a guide bushing to keep the drill from wandering off even more than has happened already. You cannot successfully start a carbide drill on a sloped surface in hardened material.
Wear safety goggles!