Agree with all the above comments.
Holidays need to be agreed between the two of you (or three of you including DP) and agreement means you have the option to say no.
She is entitled to take holiday at some point - if you have set out any sort of contract it will be in there - if you have used an agency there will be guidance on their website. It doesn't have to be the same time you are taking a holiday usually, but it has to be reasonable. That said, if she is not going to use the holiday to go and visit her family or similar then there's no reason why it shouldn't be while you take your holiday too.
If you don't want her to take holiday after just 4 weeks then you should say so - explain to her that she will be allowed holiday but not just yet. Maybe make some suggestions as to when.
What are your arrangements about weekends? Our au pairs usually have weekends off unless we ask them to babysit over the top of usual hours. I ask because you complain she is spending the weekend in London.
Agree that it is up to your brother if he is happy with her staying or not.
You don't have to allow her friend to stay unless you are comfortable with it. This is your home, after all. Perhaps you could say, if this is agreeable to you, that you will consider allowing friends to stay once you have got to know them a bit. Is the friend an au pair too? In that case they may be CRB checked too, but otherwise you should be careful.
You definitely need to be able to say no to her or she will take advantage.
I agree with above suggestion that DP get involved if at all possible.
On the other hand, if you feel that you cannot say no to her or manage her yourself perhaps you should consider ending the placement.