I think you did the right thing telling her to stop. It's not like you started a fist fight or anything. Her "cry and go home" tactic sounds like a way for her to get sympathy. But she left work without telling anyone. That could cause her a lot of trouble.

On your e-mail just make bullet points.

-She does not assist with the work and instead stands by telling us what to do
-She complains about tasks given to us, stating she is unable to perform necessary duties
Like that. you don't need to be a good writer. Just the facts will do.