Henry is King John's young son. We don't get to hang out with him until the very last scene of the play, when Shakespeare pulls him out of a magic hat after King John gets sick. Prince Henry shows himself to be of a sensitive, caring nature, and he's amazed at the brevity of human life. He seems like a pretty good kid.
Does this make us feel like England is in good hands at the end of the play? Probably not. Henry is a kid, first of all. Second of all, the play as a whole has shown us that thoroughly good people don't often fare so well in the political world.