Well you could just give her the add'l week off unpaid, but if it would be a hardship for you to find temporary childcare for those two weeks, then you may have to let her go.
I would hope that you could secure back-up care vs. having to issue her walking papers.

How old is your child OP?
Because if he is young enough still that he needs to be carried often, you do not want to compromise his quality of care.

You may have to give her a leave of absence until she is done revovering.
Without pay of course.

This is the kind of thing I tend to be as liberal as I can afford to be with. Two weeks of vacation plus 5 sick days per year is really not much time in order to take care of oneself, one's family obligations, one's health, etc.

With that being said, I am not sure how I would feel about paying for 2 weeks of extra leave BEFORE she had used her 2-week vacation.

How long has she been with you?

Obviously she can't go on vacation now. That sucks, but shit happens.

How old is DD? Does she need to be picked up and carried during the day? If so, your nanny will not be able to care for her until the doctor gives her clearance. Her one week of sick leave will cover this week of recovery. You need to talk with her about how she wishes to handle the rest of her time off. My guess is that she will want to go on her vacation since she will be visiting her parents and has paid for her ticket. The rest will have to be leave without pay.

Fire her because she is putting her interests/needs/wants/desires before yours.

Unpaid leave. Let her know she can use her annual leave instead if she wants. Her choice if she wants to still go on vacation or not.

How long has she been with you? If you love her and want to keep her, consider paying something. You could do unpaid. It is in the contract but for long-term nanny, I would consider paying partial at least.

Assuming you want to retain her after she is through all of this, you give her leave without pay and you hire a temporary nanny using an agency. (This is what we did in similar circumstances.)

Depending on the age of your child, and how important lifting/carrying is, you might consider telling her to just plan to be out from whenever she has the surgery until she is back from vacation. Pay her for the portion of that time that uses up her accrued leave and the rest is leave without pay.

You could opt to pay her some portion of her salary for the time she will be out - but that is not required certainly. It's the kind of really considerate thing that can help retain a terrific nanny - but that's very much dependent on the people and the particulars.

DD's Nanny had to have surgery, and will be out for probably about 2 weeks. Also, once she gets back, she will not be able to life DD for a while. She was scheduled to take 2 week vacation beginning of August to go visit her parents, and had paid for the ticket. Her contract has 2 weeks off, and 1 week sick leave. How do we handle this?