My young assistant is acting inappropriately...should I fire her?

Is there anyone else in the chain of command you can get to talk to her about professional expectations and behavior before letting her go? If not, then I guess you're going to have to have that conversation and fire her if she doesn't respond. Be really, really careful, though: she could very easily try to spin the story against you out of spite. You might even want to talk to an employment lawyer about how to protect yourself before you take action.