Yes, I believe it should technically be "The king was wise to know to whom the horse belonged.", but in American English, we often say "who" instead of "whom". In fact, I hardly ever hear any other Americans say "whom" in spoken language, and the only place I really expect to hear or see it is in formal writing. Most people here in Massachusetts would also ask "Do you know who this purse belongs to?", not "Do you know to whom this purse belongs?" They do this even at work, so I'm not sure if it is necessarily a formal / informal setting thing. Note that we also move the "to" to the end of the question. Perhaps another native speaker could address this phenomena. Hmmm.

The king was wise to know to whom the horse belonged.