Not to question, but wouldn't the plastic shavings from the drill end up in the engine? I'd suggest, if the piece still in there has jagged edges for the ratcheting mechanism to work, open a pair of needlenose pliers in there, so you can get a grip on the outside of the circle, and twist it out.
The cap will do this when the engine it hot, my guess is that you shut it off and then did the oil change. Not always the case, they just wear out, but I had the same problem where I could not get it to loosen, kept click click clicking, once the engine cooled down, came out fine.
If you NEED to get oil in the engine to drive it, you can pull the front breather elbow on the top of the engine, and use that hole. Be careful with the little plastic elbow, they can get brittle.