Salary could depend on how experienced someone is and what that particular individual needed to earn to pay their bills. A live-out nanny will have a lot of bills to pay and will probably live quite near you, so will be paying housing costs in your area which may well be quite high given you are in London.
Work out what you can afford to pay, remembering to include as many costs as possible. Try to end up with a figure for Salary, figure for Employers NI and associated employment costs, figure for other costs like activities and travel.
Then deduct a bit from the salary figure... so you have room for giving a payrise in the future.
Then advertise the job at that salary figure and see who applies.
Some things to keep in mind:
A part-time employee is just like any other employee and is entitled to statutory paid holiday. So calculate your costs as Yearly, not on a per-hour basis, so you factor in paying them whilst they are not physically working for you.
A part-time employee will still often have another job, so you will need to operate PAYE regardless of the salary paid.
If you are paying enough, you will need to also pay Employers National Insurance.
Mr Anchovy's PAYE Calculator is useful for playing with figures to see how salary paid affects Employers NI. It is also useful for seeing how TaxCode affects the nannies take home pay. A part-time nanny could have taxcode BR (meaning that all personal tax allowance is used in another job).
DO NOT AGREE A NET WAGE
To get a feel for salaries on offer, look on various nanny recruitment websites. This Search on NannyJob will give you adverts for London, Part-Time, Live-Out. Some of the ads may mention salary, though you may need to the the PAYE calculator to convert Net to Gross (remember to set the taxcode to BR, so you get result for worst case situation where all their personal tax allowance is used in another job).
It would not surprise me if you are looking in the £12-£14 Gross per hour range for an experienced nanny to work part-time, live-out in London. However as an employer you can offer what you like (has to be at least National Minimum Wage, currently £6.04 per hour for someone aged 21+) and see what applicants you get.