There was the following sentence in Maureen Dowd’s article titled “Taxing Times for Obama” in the New York Times May 18 issue. - http://www.nytimes.com/2013/05/19/opinion/sunday/dowd-irs-investigation-means-more-taxing-times-for-obama.html?hp&_r=0

“Asked about that on Thursday, Obama might have tried a little J.F.K. wit to dismiss the ridiculous assertion. Instead, he played the pill, as he too often does, huffily telling reporters, “Well, I’ll let you guys engage in those comparisons, and you can go ahead and read the history, I think, and draw your own conclusions.”

What does “play the pill” mean? I was unable to find this expression as an idiom in none of Cambridge, Oxford and Merriam Webster Dictionary, nor in Google N Gram.

At the same time, what does ‘J.F.K. wit’ mean? What is it like for example?