Well - you have options... :)
Some people advertise for 'split shift' or 'wrap around care' - it's possible to get a nanny to do that, but they are also likely to take another job when it comes up.
You could do as you suggested and offer to pay half - but only if you don't expect her to do anything at your house during those hours.
Or you could do what most people do and pay her for all the hours. She could then prepare meals for the children to put in the freezer (for you to use on other days & her to use on her days), do the childrens' laundry, keep their rooms clean & tidy, change their beds etc.
If you only pay her half or nothing, you can't expect her to have them if they are ill or for any other reason they can't go to nursery.
Could you do drop offs and just employ an afternoon nanny?
When is your eldest going to school? Remember you will need to sort out holiday care then too.
What about a childminder instead - it's cheaper.