What factors motivate Kit to dive into the water to retrieve Prudence's doll?

Kit was motivated by the fact that the child was crying, the mother was angry, and the Captain wasn't paying attention to any of it..... more than that, he ignored Kit's pleas. Kit was angry.

"Turn back, Captain," she ordered impulsively. " 'Twill be an easy thing to catch."
The captain did not even glance in her direction. Kit was not used to being ignored, and her temper flared. When a thin whimper from the child was silenced by a vicious cuff, her anger boiled over. Without a second's deliberation she acted. Kicking off her buckled shoes and dropping the woolen cloak, she plunged headlong over the side of the boat.