I would say, mostly, what you said here.
First, I would find out why she doesn't like working alone. Is she afraid? Is there a security issue? Can you do anything to fix that or reassure her?
If she's just social and likes to have someone around, that's too bad, assuming that she knew she would work alone when she interviewed. Did you explain this part to her? It's a small business, and the reason you have her there is so that you can do whatever else you have to, right?
Sometimes small companies don't have written job descriptions. If you don't, can you construct one based on what you told her in the interview? You can find templates and examples on line. Write it all down. Then sit with her and say that you want to be sure she agrees with what's on the list. Have you left anything out that you have asked her to do or that she has added to her task list which you find beneficial? That makes it collaborative, but you're still the boss. Is there anything she is doing that you don't want done or need done?
Do you have any kind of written personnel policy regarding vacation, sick, and personal time off? If not, write those up too. Even if you have them, write them up. You can indicate your position on being "flexible" such as how much notice she is supposed to give and what format her requests should take. You write these as if they applied to anyone you employ, not just her. If she's just asking for time off and you are scrambling to cover her job, that needs to stop.
Then you review her performance: you indicate to her the things that make her work, in your words, "outstanding." Then you list areas for development, growth and improvement. Professional demeanor does not include nagging. She can register a formal complain about something if she wants to, but she can't complain about a job description and requirements she agreed to. If an employee wants to suggest ways to broaden a job or make the work environment more efficient or pleasant, that's one thing. But complaining that the policies and environment she agreed to aren't to her liking when this is the job (!) isn't one of them. You can tell her that it's a negative on her performance evaluation and something you would like to see changed because it's unprofessional and a nuisance. If this isn't the job for her, then so be it. You can agree to have her leave and you can find a replacement.
Sometimes small businesses set things up in such a casual manner, employees act like they are family members or friends rather than staff. You can all upgrade your professionalism to get rid of these problems. Ultimately, you are the boss.