The problem with doing reverse calculation is that it can come out differently.
Example:
40 hours per week, 5 days per week, no student loan
Taxcode: 1000L
6.50 net = 7.47 gross
Taxcode: BR
6.50 net = 8.89 gross
Taxcode: 500L
6.50 net = 8.18 gross
If it was 6.50 gross, then you need to be careful about National Minimum Wage which will rise to 6.50 gross for those aged 21+ from October 2014. Source. So ok for now but may not be if NMW rises again.
Talk to them, find out if the 6.50 is what they want take home or if they do mean gross. If it is take home, then decide what you feel able to offer... 7.50, 8.00, more... and come to an agreement on the gross salary.
If they insist on having a set net wage then you will have to use a nanny payroll company. Talk to the payroll company before confirming with the nanny. I would not suggest you agree to paying a Net wage as you do not know for certain what their taxcode will be, especially if they have other work.
As others have mentioned, the NWOC with two children may well be finding it hard getting work as the job needs to fit around her own children's care. You will need to decide if having both the children coming to your home during school holidays would work for you, plus the daily travelling to/from school, or what arrangements you would be expecting the nanny to have for before/after school care of their eldest child.
It could work out well for you and for them, so think about the salary you will offer and all that may be involved given each applicants circumstances.