In K-means, what happens if a centroid is never the closest to any point?
You should leave the lone centroid unmoved. In the next iteration, it's possible that the cluster centers have moved in such a way that now there are some instances that are closest to the lone centroid, and it can get picked up. At the end of the k-means algorithm, you could remove clusters that have no instances associated with them, but it doesn't really matter.

At the end of the day, it's probably not a good idea to use k-means in the manner that you described, and we should not expect it necessarily to produce good results.