It appears this is the same location as Honolulu Harry's Waikiki but that the name might have just been shortened to "Club Waikiki" in later years. Matchbooks with the same address point in that direction.
Alternately, photos of Harry's show "Club Waikiki" painted on the side, so it's possible this club was just a section of the larger complex like Don the Beachcomber's had its Dagger Bar, for instance.
Today, the site is home to The Covington Luxury Apartments, which were built in 1965, so Club Waikiki must have closed some time between 1959 when Harry's was doing dance reviews and dinner theater and 1965 when the Covington was built.