Nursery nurses are usually pretty clued up about Gross wages, as nurseries pay Gross. So I don't know why she is asking for a Net wage.
Please avoid using lookup tables in this situation. As you have correctly identified your nanny is/may have another job.
You don't know what her tax code will be at any given point
That is true for ALL jobs. While a person with just one job could be assumed to have a tax code currently of 647L, it may not be the case.
So... in this case, use a Net to Gross calculator to work out the Worse Case scenario. Which is taxcode: BR - which means the entire lot is taxed.
She wants: £10 x 12 = £120 per week. Enter in £120 in the Net per Week box.
Enter in Days: 2 Hours per week: 12
Taxcode: BR
Click Recalculate.
Currently, it calculates at £680 Gross per month. Employers NI is £26 per month.
In April, tax rates change. So the figures will be different then. They can also vary slightly month to month when a payroll company aims to get the same Net figure.
So... you need to agree a Gross wage with your nanny. Not NET. Then it keeps it as simple as possible and any payroll company can deal with it. Not sure why you want to use your employers payroll company specifically - assuming they are charging you a fee for using them, I'd suggest shopping around for a payroll service.
So, if you are wanting a Gross Per Hour figure, then £13 Gross per hour is closeish to £10 Net in this situation.
£13 gross as we have discussed on a recent message thread, is quite HIGH for a nanny. Though you are only wanting part time hours, so that does tend to come at a bit of a premium. Are you London based?
Oh, YES you will need to operate PAYE as per how you did for your nanny on ML. Reason being: the amount being paid is above the earning limit. It is also their second job (have they completed a P46 to inform you that it is their second job?)