Upvote:5
In general, good contemporary sources for questions like this one include city directories and old maps. There are multiple sources of maps, and I prefer to try to access city directories at FamilySearch because it's free.
In 1908, the Kenwood area of Chicago was likely served by a station that was on present-day S. Lake Park Avenue between E. 52nd Street and E. 53rd Street.
According to the 1902 and 1908 Lakeside City Directories for Chicago, the Chicago Police Department was apparently organized into Divisions 1-5, Districts 1-13, and Precincts 1-44 (1902) or 45 (1908).
These directories list what was then the 4th District, 10th Precinct at 5233 Lake Avenue. This was also the 2nd Division headquarters.
Blanchard's map of Chicago with the new street names: 1908 shows that Lake Avenue was the former name for what is now S. Lake Park Avenue. Looking at historicaerials.com, it seems there used to be enough space for buildings between the east side of Lake Avenue and the railroad lines, but apparently the road was rerouted between 1963 and 1972, and those buildings were lost.
Across the street from the location where I think the 10th Precinct station used to be are now an LA Fitness and a Starbucks.
1902 directory listing police stations:
1908 directory listing police stations:
1908 map of Kenwood neighborhood:
1963 aerial view from historicaerials.com:
1972 aerial view from historicaerials.com:
2009 aerial view from historicaerials.com:
2022 Google maps: