Since you are her boss, I believe I would give her a written warning about her behaviour and lack of responsibility and then if she continues, fire her.
Chaos breeds chaos in offices, and this isn't going to get better as long as she's around. Going into cabinets that she knows she should not be in (key or no key), pilfering through anyone's desk and bad mouthing her boss (you) are intolerable.
Second thought, forget the warning and just fire her. She'll figure it out all by herself.
Hugs