Jazzbaby appeared in Astro City during the same time frame that jazz music took form and became a prominent part of American popular music. The precise date is unknown, but can reasonably be estimated to be sometime between 6 April 1917 and 1919 - America's entry into World War I, and a known interaction with Dame Progress.
If the The Broken Man's telling is to be trusted, she is a later incarnation of Mister Cakewalk. This would suggest that she is also connected with The Bouncing Beatnik and The Halcyon Hippie, and that the connection between the four characters is the music of the times. There may also be a connection to the destruction of Silverstring's guitar.