Helicoils come out naturally, but a lot of the suggestions above make the problem worse. An ez-out for example pushes outwards due to its taper, and that will make the helicoil tighter, pushing it deeper into the thread's grooves. A left handed drill bit has nothing to catch on.
You shouldn't need to do all that drilling damage. A fine dental pick will get you started enough to drive something bigger under the exposed tail. Grab that tail with pliers and bend it in, towards the center of the hole. Then just rotate it, and it will release and unscrew, just like the bottom tail of the helicoil works with the insertion tool.