"She charges £9 an hour" - no she does not. You are the Employer. You decide how much to pay. You offer the job, she decides to accept it or not.
If she is wanting £9 an hour, is that take home (Net) or Gross? Net to Gross is not an exact science:
With taxcode 1000L (tax year 2014/15) and nothing special, then
If they work 40 hours a week - £9 net = £11.15 gross
If they work 50 hour a week - £9 net = £11.57 gross
Net to Gross does not really work as the Gross amount will change depending on number of hours per week.
So as you are in London I would guess that £9 is not the Gross amount she wants. I suggest you come up with a gross figure and settle on that. £11.20 perhaps, £11.50. YOU decide what to offer, let her decide if she will accept it or not. Do NOT give any guarantee of a take home wage amount.
Contact a payroll company and see what the current procedure is for setting you up as an employer. They may be able to do that for you, though you may need to activate things such as GovGateway, HMRC Online. Talk to a payroll company.
As has already been mentioned, the process once underway is:
Monthly payslips
You pay the nanny the net amount shown on the payslip
You put aside the deductions shown on the payslip (employee NI, employee income tax, employer NI).
Once every 3 months, you pay the money you put aside to HMRC using a specific payment account (your payroll provider will remind you of when to do this, the account number, sort code, reference number to use).
Once a year the payroll company will ask you about any other payments made to your nanny, such as for mileage (you will also be asked number of miles).
The payroll company completes the Employer Return (P35). They will issue a P60 to your employee (this is usually in May) and will issue a P45 when your employee leaves your employment.
If your nanny will not agree a Gross salary, then you MUST use a payroll company.
If you have the time and feel the effort is worth it, you can do payroll yourself as long as the nanny has agreed a Gross salary. You use the HMRC Online system plus writing out payslips manually, or you can use software like Brightpay.
Nanny payroll companies vary in price and what they offer - all will offer the basic service of producing Monthly payslips and doing electronic filing of documents to HMRC.