Does the photographer's inscription say "Dakota" or did it say "S Dak" or any other variation of South Dakota?
A couple years ago when I searched for photographers in the Brookings area, I found a directory that said in 1890 Ole G Oyloe was a photographer.
However, if the inscription just said "Dakota" that would likely indicate it is from before 1890. South Dakota
became a state on November 2, 1889.
I also read that Ole Oyloe bought out a photographer named Jarvis in 1891. Then in later years, John G Oyloe and Conrad Oyloe were also photographers.
I have several photos from the Brookings area and quite a few were taken by the Oyloes. The inscriptions I have seen all say "Brookings, SD" or "Brookings, S. Dak" though.