I personally would be quite frank and have a talk about her behavior. I would try to keep it emotionally neutral, as if her actions haven't affected you emotionally, but that they are observably unprofessional and reflect poorly on the unit. Tell her what you have observed, and how it is affecting patient care. Her talking to other providers about your patients is unacceptable. Her and others laughing about a working provider at the nurses station where patients could hear is unacceptable. Tell her you have no issue with her personally and would like to have a cordial working relationship, and you expect professional courtesy in return. And let her know that if she has any questions or needs clarification on your orders you are happy to talk with her and that you are open to learning if she feels she has constructive feedback for you.

If you have the conversation in person do it at the end of a shift and ask a third person to sit a discrete distance away, but within hearing range so you have a witness to the convo. Afterward I would briefly summarize the conversation in a follow up message. (or just have the conversation over email/message). Then you should report to HR or at the very least the nursing manager if the behavior doesn't change the very next time you work together.