Have we at last reached the point where comedy routines have their own (carefully analyzed, but quite unfunny) Wikipedia articles? Yes. Yes, we have.
The Three Stooges do it as performers-within-a-performance in Gents without Cents. This is the version I remember from my childhood, and it has the best-timed violence.
The best performance I have found is Frank J. Scanell, as a comic hired to teach "a burlesque act" to Lucille Ball. It startes at about 0:50 in this clip, and runs to 6:00.
Lou Costello does it with Sid Fields. Somewhat sloppy.
An aged Moe Howard reprises the old routine with Mike Douglas(!) in 1973. Extremely sloppy. The novelty of cameos by Roger Miller and
Another thought: This routine must surely be detested by anyone who must tell people that he or she is from Niagara Falls...