As a nanny/housekeeper myself I'd be pretty cross if my employer went to the agency and not myself.
I've always, as a nanny become part of the family unit and we work together.
Here's some ideas.....
Personally over the years I've found that mother help jobs or jobs when a parent works from home can be rather challenging.
Either feeling micromanaged (even if it wasn't intentional) or the children behave differently because they don't know who's boundaries to follow.
Have you gone for someone qualified in childcare, with experience, paediatric first aid etc? Maybe she isn't confident with children if she expected to be mothers help or a house keeper predominantly.
The wages are fabulous and she should appreciate that (or maybe I should move to the city) working a 7-7 for 5 days that's £15p/h which is about right.
Equally the night out thing if it isn't specific days like every first Saturday of the month in the contract then it doesn't work because maybe she feels like you own her as such. She is entitled to have plans and even work another job.
Nannying works both way, you have to get on with the family and the nanny and work as a team for a stable upbringing of the children. She may be brilliant with them but the kids will pick up on the atmosphere.
It sounds like nannies are like gold dust in your area but you have to go off the vibes and almost become friends who can take guidance on jobs. I've always had a family for 8-10 years and that's ideally what you want. A extended family member who works their socks off.
Maybe she needs guidance, is she young? New to the work?
I'd also explain it's a large outgoing for yourselves so you have to ensure she's using the time effectively. She may feel awkward you're around.
Give her set times- structure.
Example every Tuesday and Thursday you have the baby sole charge 9-12, baby naps 12-2 so housework can be done in this time and then you do the school run.
Spreadsheet your needs and have a 1-1 chat. Tell the agency not to contact her.
I hope this helps