Which Kansas City does Wilbert Harrison speak of in the song "Going to Kansas City," where he sings "got some crazy little women there and I'm gonna git me one?"

Kansas City, Missouri

In 2005, Kansas City adopted "Kansas City" as its official song, dedicating "Goin' To Kansas City Plaza" in the historic 18th and Vine Jazz district. The Beatles' version of Kansas City is played over Kauffman Stadium's sound system after each Kansas City Royals win, while the Wilbert Harrison version is played after each loss.

The lyrics of the song mention the intersection of "12th Street and Vine", a location now marked in a park in Missouri.

https://en.wikipedia.org/wiki/Kansas_City_(Jerry_Leiber_and_Mike_Stoller_song)

The City Council of Kansas City, Mo., voted in 2005 to adopt "Goin' to Kansas City," as the city's official song. You can retrace the history of the song's famous intersection at a new park at 12th & Vine streets. The Goin' to Kansas City Plaza at Twelfth Street and Vine is a piano-shaped park with interpretive displays about the area's heyday as an entertainment district when jazz clubs and gambling halls lined 12th Street. Visitors can pose for pictures under a replica of a historic 12th Street & Vine sign.

https://en.wikipedia.org/wiki/Kansas_city
http://www.bluesforpeace.com/lyrics/kansas-city.htm

Jerry Leiber and Mike Stoller wrote the song in 1952 as a tribute to Big Joe Turner, who was from Kansas City MO - so I would assume that is the intended reference.

https://en.wikipedia.org/wiki/Kansas_City_(Leiber_and_Stoller_song)#Original_song

There is no corner of 12th Street and Vine in that city, but there once was, before urban redevelopment during the 1960s. And it was the centre of action!

http://www.kcur.org/post/news-flash-world-kansas-city-has-no-12th-street-and-vine-heres-why