The footage does, indeed, exist, as a TV kinescope. Gertrude Lawrence sang both "I Whistle a Happy Tune" and "Getting to Know You" in costume, with Richard Rodgers accompanying on the piano, as the climax of Part 2 of Ed Sullivan's Toast of the Town tribute show to Oscar Hammerstein II. Broadcast date was Sept. 16, 1951. Alas, there was a problem with the video for "Tune," so most of that is audio only. But for "Getting to Know You" the video problem was fixed, so we have all of that performance.