The simple way to figure this out is, as Michael said in a comment to another answer, to drop the first part of the combination:
Would it be
After all, what better way to have the last laugh than to make I say nice things about him to a national audience.
After all, what better way to have the last laugh than to make me say nice things about him to a national audience.
This makes it obvious that Obama made a mistake. And that may seem strange, since he is not only a native speaker of English, but he is quite an accomplished user of the language, and an example to many.
So why would he make such a mistake? I think what's going on is a phenomenon called hypercorrection. For quite a while, speakers used to use me in sentences like the one in this question, even when it should be I, resulting in sentences like
*George and me are off to the super market.
*Paul and me failed our exam.
Teachers and other people who insisted that I should be used in these sentences seem to have managed to convince people that the use of me is always wrong, resulting in people overcorrecting and using *he gave John an I a present.