With nannies (part-time ones especially), always agree a Gross Wage in the contract. Should there then be changes to the nannies tax code, you do not get a nasty shock.
Nanny agencies often advertise pay rates as NET. However, an increasing number of agencies are now giving Gross figures and I hope to see that increase over time.
To calculate how much employers NICs are likely to cost you, use an online PAYE calculator such as E-Gismos UK Pay Calculator which will break down a gross salary into Income Tax, Employees NI and Employers NICS. You can find out more about your tax responsibilities as an employer in the HMRC New Employers Pack. The P49 - Paying someone for the first time booklet is worth a read as it takes you through your first payroll run.
The tax/ni side of things with one employee is not difficult. Yes it takes time, but HMRC make it as easy as possible, providing a CD-ROM for Windows PCs which does a lot of the work for you. They also provide a new employers helpline should you get stuck or just need reassurance that you are doing things right. As others have mentioned, you can also use an agency to do your payroll for you.
If you have agreed a net wage of £9 per hour, for say 30 hours per week that is £270 per week, rougly £348 gross I used Listen To Taxman to calculate this figure
Which is around £18100 per year, so agree a Gross Figure of a figure similar to this in the contract. Using e-gismos calculator, Employers NICs is around £1673 per year.
Based on 30 hours per week, 52 week year, this is 1560 hours per year. (£18100 + £1673) / 1560 is around £12.68 per hour.
So yes, your friend is correct... if you have agreed £9 net per hour, the real cost to you is over £12 per hour. If £9 per hour is Gross then of course the figures are all a lot a different.
With 2 children, a childminder in my view is often a little cheaper than a nanny. However there can be disadvantages, such as a childminders opening hours and their sickness policy (nannies tend in my experience to be more willing to care for a child with minor illness, whereas childminders have to protect the other children in their care).
You will find a lot of information in this Mumsnet section regarding Childminders and Nannies which should help you understand the differences and thus help you decide which form of childcare would work best for your family.