I never let kids walk around in my pasture without an adult. I have seen little kids injured that way.
It sounds like you are on the right track. About using the whip, don't just scare her; sting her. You don't want to injure her, but unfortunately it sometimes requires actually popping them to get your point across. You won't have to do it often though you probably should still carry the whip around with you.
One nice thing about horses is that they are quick to forgive you if you correct their bad behavior. They do not feel secure in life if they don't know where they belong in the social dominance hierarchy.