The question particle kai is used at the end of sentences in place of the ka question particle.
Kai is generally used by men for questions or doubts to their wives or subordinates (not to superiors), including children.
It is probably a contraction of -ka-yo and it is related to dai. However, Hayaku kaeru-kai? "Are we going home soon?" isn't as argumentative as Hayaku kaeru-ka-yo?