I would say Howard County has multiple high schools (throw Eastern (Greentown) in there as well), not the city of Kokomo. Although Taylor is close. Residents of the Northwestern and Western school districts want nothing to with the city itself, other than a job a Chrysler/Stellantis transmission plant.
The city of Kokomo did once have two high schools. Haworth High school was open from 1968 to 1984, and the current Kokomo high school building on the southwest side was originally the Haworth building. IIRC Haworth had it's own sport facilities except for basketball where both schools shared historic Memorial gymnasium, a decision that rankled a lot of Kokomo fans at the time.