I think some of the comments here are a bit harsh. Whilst the advice is correct, OP admits she is new and is looking for advice. Rather than criticise her for the oversight (which she has already admitted) it would be more productive to point her in the right direction.
Here's my advice:
1- Get yourself a contract pronto. (you can cat me for a sample one if you like)
2- Yes she is entitled to 28 days, including bank hols. This is prorated if she is less than 5 full days per week. 4 days per week works out to a total of 22.4 days.
3- I suggesst you put SSP only in the contract. The norm is pay her anyway if she is genuinely ill and use this as a fall back position if she is either taking the mick or her sickness time exceeds your budget.
4- Outline discipline and dismissal in great detail. List what you feel should qualify for gross misconduct and therefore immediate Search MN threads for lots of advice.
5- Consider house rules. (mine are attachment to contract but they don't have to be)
6- nannypaye (and some others which I can't think of just now) are cheaper than nannytax.
7- Regarding holiday, many people allow their nanny to choose half of them. This is not a legal entitlement. Legally, she/he requests days and you have the right to approve/decline them. But, of course. you do have to give opportunity for them to be taken. So, you could just tell her the snow day is unpaid or you could give her the option of taking it as hol if she wants to. She does not have a right to take it as holiday as she has not requested it. And, speaking of requesting holiday, I would clearly outline in the contract how much notice you expect on a holiday request and if you want it in writing. In my house, I ask the nanny to take her hols in term time because I don't fancy taking hols from my own job just to do the school run. But, if she has a reason she needs certain dates (i.e. frineds wedding in another country), DH and I will bend over backwards to accommodate.
I hope this is helpful!