I have just been through this.
Firstly, you need to register as an employer - use the form PREM REG at this website: Revenue
You then need to register for ROS Online (you can do all your payroll via paper but it is surprisingly easy to do it online). The instructions are Here.
When you get your information for ROS Online (it takes a few days), log in and go to 'My Services'. Then go into 'Complete a Form Online', click 'New Employee' and enter the information from the nanny's P45. You should, within a couple of days, get a file in ROS Online which gives you her tax credit information for payroll purposes (am not 100% sure of this because mine began at the start of the year so got a tax credit cert as a matter of course).
You will need a simple payroll package. I am using SimplePay. There are a few out there but stay away from SAGE - I used it initially and it was a nightmare to deal with and also made a serious error in the first payroll calculation.
I'm happy to be of help if needed - PM or just message here.
Some of the nanny agencies will do payroll. I know the Belgrave Agency in Dun Laoghaire offers this service but I did not end up hiring through them so didn't avail of it. Once you get set up on payroll though it's actually easy enough to do it yourself.