Answer:
He
Step-by-step explanation:
"He" is a subject pronoun, and "Him" is an object pronoun, in order to understand why one fits in the sentence and the other does not, it is important to differentiate between these two.
Subject pronouns are the pronouns that perform the action in a sentence or the pronouns that are being discussed and deal with within the sentence. In this case, "His partner and he" are the main focus on the sentence, and they are the ones who perform the main action which is to be best dancers. (It is important to note that "He and his partner" sounds much better than "His partner and he").
As for object pronouns, they are pronouns receiving the action in the sentence.
For a better understanding of these concepts, check the following sentence:
She gave him a book.
Who does the main action (to give a book)? "She", therefore, "she" is the subject pronoun.
Who does receive the main action, or who are given to the book? "Him", therefore, "him" is the object pronoun.