I believe that "What's A Nice Kid Like You Doin' In A Place Like This" was written for an animated Television version of "Alice In Wonderland." It was broadcast sometime in the mid-60s and while I don't remember much about the show I have never forgotten the song. Sammy Davis, Jr., as the Cheshire Cat, voiced it. I had no idea Kander and Ebb had written it. UPDATE: According to IMDB, it was written by Charles Strouse and Lee Adams so maybe Kaye sang a different song with the same title?
Kaye Ballard is a treasure! I look forward to seeing this documentary.