1) My staff will do their best to make sure I enjoy in my stay.
2) My staff will do their best to make sure me enjoy in my stay.
Which pronoun should I choose after "make sure", subject or object?
After the "make sure", you need to choose a subject because it has to be:
"make sure (that)" + a complete sentence
for the whole sentence to be complete. Using your example sentences, it should be:
My staff will do their best to make sure (that) I enjoy my stay.
But to me, "my" staff making sure that "I" enjoy my stay sounds a bit odd. I think it needs be something like:
The staff at the hotel will do their best to make sure (that) I enjoy my stay there.