Iroman 2 is set only 6 months after the first movie, the surgery happened 3 years later(Stark have time to discover a way)

It's also that he wasn't emotionally ready for that. The surgery symbolized Tony's attempt to move on from that kind of life (even though he failed miserably, he instead started focusing on preparing for the next bad thing instead of fixing the current one). At the end of Iron Man 2, he's still stuck dealing with the trauma he's experienced.

"6 months after the first movie" ; more accurately, "6 months after the end of the first movie".

Didn't he use a reworked extremis formula for the surgery? I could have sworn that was heavily implied.

as far as I recall, the details for both him and Pepper were vague.

about why not in second Iron Man; some other possibilities::

- he possibly not yet figured it out by then.

- the technology possibly not around by then.