Minki
Take a deep breath.
'Stupid' who knows - she's unlikely to have told them they are stupid and much more likely to have told them to stop being stupid. If your children are, as you say, naughty - then a nanny they don't like might not mean the nanny is the wrong one! She might just be expecting decent behaviour from them and that's no bad thing.
You need to talk to her and explain that you cannot take time off at short notice or arrange other care for the children so she needs to make other arrangements for her appointments.
I often have to work late and so will be back from work late, often with very little notice
You are employing her from 3pm to 7pm. If you need a nanny to be available, at the drop of a hat, for longer hours, you need to be paying her a premium for never knowing if she will be working 'late' or not. She is as entitled to have a life outside of work as anyone else is. She's not 'on call' unless you pay for 'on call' services AND the extra hours on top of that.
If the children go to their Dad's or anywhere else, you still need to pay your nanny & NOT expect her to work the time in lieu. She is available for the contracted hours, if you choose not to use them, that's up to you. (I realise this week was unusal - but I'm just saying - going forward).
Also 'I will need more hours in the holidays' - well, if she's available and willing, great. If she isn't you will have to find an alternative. You simply cannot have all the perks of a full time nanny and only pay a few hours after school.
As for tidying?! I imagine she has plenty to do with your 'lively' children and all the afterschool stuff without doing much in the way of housework. She should be leaving it much as she finds it - but you need to be honest about how much is reasonable with your kids.
You are trying to have your cake and eat it too I'm afraid.