You don't agree to NET wage. The amount you pay her, could easily at any point, especially at each new Budget. As the employer, you lose out if you agree to paying a NET wage. Please don't agree to that.
Agree a Gross wage. Make sure your nanny understands that the amount they take-home will change for various reasons though-out their employment, such as if their tax code is changed, if tax rates change etc.
You can use ListenToTaxman to calculate roughly what the net wage, from a gross wage will be. So if your nanny wanted £100 per week net, and her tax code in your employment was BR (thus not taking into account any of her personal tax allowance - as this is likely to be used by the other family), then gross pay is around £125.15 per week.
When she starts with you, you will need her to complete a P46 and from that, you will get the starting tax code. HMRC will then inform you if they wish you to use anything different (such as if the nanny has requested to them, that she wants her tax code split between her employments, or if HMRC are clawing back any back-tax due, or any other payments taken directly from source).
If paying 125.15 per week, that's £6507.80 per year (52 weeks), using E-Gismos UKPAY, with tax code 0 (BR) then it says that Employers NI is £164.20 per year. Please keep in mind that these calculators may not be precise, as there are many things that can affect tax calculations... but they are a pretty good guide.
The employers NICs is calculated on the amount you pay your employee. Other employments do not come into the equation.
To make things easier for yourself, forget the other employer exists - as it isn't your problem. All you need to remember is that you always pay GROSS and that your nanny needs to complete a P46 (as she won't be giving you a P45).