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 any of Cambridge, Oxford and Merriam Webster Dictionary, nor the incidence in Google N Gram.

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