Thank you for your question.
We have come to rely on the Internet for near perfect knowledge; it is as close to that as we can be. The problem is, that local information grows obsolete quickly.
This does not mean your search (or ours) is in vain, it means we have to change how we search.
The information for a local band and its songs, which have not entered the mainstream will not be found in the Internet space. there are several versions of a song, for example called caution, but by Bob Marley, and others, who are main stream and public figures. And as sometimes happens, different songs carry the same name, so the songs I found are not the ones associated with your band.
so for now you would need to go to the local public library and search the name of the band amongst the news paper archives.
You can also call on disc jockeys of the time who are operating in Wichita, Kansas.
Most local disc jockeys and music radio station hosts in Wichita would be able to provide that information, or at least give leads.
If you put your detective hat on, and find a newspaper article about them in the local archives at the library, you can get a name. The song is most likely copyrighted, and you can search on it at the national archives. You will not only need a name of the band, but it would also be helpful to know the name of the members of the band, because the copyright, may have been in the writer's name rather than the band's name.
If you are not able to find it in the on line archives search, you can wait until office hours when there is a librarian available, and ask them to see if they can find it. There may be a small service charge for a so called assisted search.