Answer:
"Her"
Step-by-step explanation:
A princess refers to a girl or a woman ruler, meaning that the most suitable pronoun that refers to it has to be of a female. "Her" is the pronoun that refers to the third person singular, and it used for females, thus this is the correct one.
If we chose "his" to complete the bank, it would indicate that the Prince is the one that owns the state, and to complete the bank with either "his and her" or "their" would indicate that the state belongs to both of them and not to the princess only.