I checked a few videos claiming to cover the pronunciation and the first two said it more like "kissy me." But this one has the pronunciation I'm familiar with:
https://www.youtube.com/watch?v=Nu-Kq-2irQE
Edit:
The name Kissimmee can be traced back to the language of the Jororo people and means “long water.” There were approximately 350,000 people living in Florida when the Spanish arrived in 1513.