Houdini was portrayed by Alan Alan, the English escape artist who died in 2014. Alan did his signature upside down chain escape from a burning rope. According to a user on IMDb, Alan's act was cut down from 13 minutes to six minutes to eliminate the comedic elements and make it a more straightforward Houdini escape.
Other celebrities who portrayed Palace stars of the past were: Debbie Allen as Josephine Baker, David Cassidy as George M. Cohan, Gregory Hines as Bill Robinson, Ann Jillian as Mae West, Rich Little as Jack Benny, Dorothy Loudon as Fanny Brice, Jeanne Moreau as Sarah Bernhardt, James Whitmore as Will Rogers, and Shelly Winters as Sophie Tucker.
You can read a New York Times review of the show HERE. You can also view a short clip on YouTube.
For fun, here's a photo of the Palace with the real Houdini headlining (also notice the real Fanny Brice on the bill).
Thanks to Mark Willoughby for sharing the art and bringing this one to my attention.