Nannies are a very costly option for the care of one child. Your nanny could well be doing some child related duties, especially babies laundry, tidying bedroom, wiping down surfaces in kitchen, loading/unloading dishwasher etc.
Are the half-days exactly half-days? Given that those days are a different number of hours to a full day, I would suggest working out holiday entitlement in Hours. That way if nanny takes off a full day then it's x number of hours of holiday taken. If they take off one of the half-days, then it's whatever number of hours that is that gets deducted.
You say you have agreed to splitting requests 50:50 but you have authorised 17 days worth of holiday of which only 5 days you requested, so 12 requested by nanny. If holiday entitlement is 22 days (see above about calculating it in hours given the half days) then 11 days would be half... so you have let you nanny decide on more than half.
"The contract says that any unworked days has to paid back to employer."
That does not sound like a usual term found in a contract... or do others disagree and feel that term is often found in a contract?
What do you mean by that term exactly? I would have thought that once nanny requests more than their holiday entitlement, you would simply decide for each request if you will permit it and pay, or permit it as unpaid leave, or not permit it.
"Nanny says any leave days above the 22 days entitlement is only unpaid if she requested them, not when I requested them."
Yes I would agree with that. If you want more holiday than your nanny is able to take off as paid leave, then those extra days should be paid as per normal. It's up to you to book those days when you need nanny to use their holiday entitlement as early in the leave period as you can. If you have left it late to book them and have authorised the nanny to take a lot of their leave already, then you could well be in a position where the nanny does not have any remaining leave to use for those days.
"She has already booked herself on a cruise, so I suppose I won't be able to ask her to come in to do other work."
You authorised that holiday request when the request was put in. I don't feel you can change it. Before authorising holiday requests you should consider what impact that has on the remaining days of holiday entitlement and your plans to go on holiday yourself.
Sounds like you have not planned your visit to grandparents that far in advance, thus are now in the situation that your nanny has not got enough remaining annual leave to cover all the days you will be away. I do not feel that is the nannies problem and not paying the overlap days would not be right.