I'm not a payroll specialist but do run a small manual payroll so have some experience! I'm happy to be contradicted by anyone with more experience though . Hope what follows helps!
You'll need to know her daily rate (or her hourly rate maybe)
Multiply her daily/hourly rate by the number of days/hours you need to pay her to get her gross wages for the bit of the month she has worked.
You will also need to pay her any holiday that she may be owed. That can be a bit more complicated! Everyone is entitled to 4 weeks holiday per year but that can include bank holidays (sorry you probably already know that, but just in case I put it in!) You'll need to know when your holiday year runs from and if she has already had any holiday yet.
For each week of the holiday year worked you are entiltled to 4/52 weeks holiday (it works out as approx 0.08). So if the holiday year starts on January 1st (most often does!) then she would have been entitled to 29x4/52 weeks holiday so far this year (that works out at about 2.2 weeks). Assuming she works 5 days a week that means she is entitled to 5x2.2 days holiday (11 days).
Now take any days she's already had as holiday from this for example 11-5 = 6
then pay number of days owed times daily rate as holiday pay outstanding on top of gross salary for this month.
Only thing left to do is write out her P45!
Hope that helps!
ps I make it 5 Bank holidays already this year so if your holidays include bank holidays then she has already had 5 days holiday (unless she worked through them!)