What happens to potential energy that is not completely converted to kinetic energy?

The concept is similar to like eating half of a meal (the potential to kinetic) and giving the rest away to your dog (the rest of the energy that wasn't converted into kinetic energy)

This ties in with the 1st law of thermodynamics, but to get straight to your answer, the rest of the potential energy could turn into heat energy as a byproduct of that starting up that kinetic energy.