Standard hols is usually 20 days plus bank holidays, half chosen by you, half by the nanny. You may want to specify holiday dates are chosen by end of March for example so you can all plan.
As per above, contractually put in standard sick pay only. Most folk I know still pay in full for the odd day but covers you if you get stuck with a long leave of absence.
I agree with making sure all tasks are covered. In mine I have (swap name for child):
(a) Looking after child's needs, welfare, development and health and safety
(b) Supervision of child and any further children at all times
(c) Babysitting at times agreed in advance
(d) Playing with the child and doing age appropriate activities, outing and classes with her as discussed with the parents
(e) Encouraging development in skills such as feeding, speaking, toilet training, dressing, telling the time, etc age appropriately
(f) Nursery duties, including, but not limited to:
a. Keeping child's bedroom, clothes, equipment and toys clean and tidy
b. Responsible for child's laundry and ironing of her clothes and bedding as required
c. Shopping for child's needs. Keeping the kitchen, breakfast and any other room clean and tidy after use
d. Preparing and cooking healthy age appropriate meals for child as required
e. Keeping the nursery and other rooms used tidy at the end of each day
f. Taking child to activities including swimming once a week and arranging play dates with her peers
g. Taking child to the doctor or any other appointments as discussed with the employers
I would suggest signing on with a payroll agency, I use nannytax but often mentioned as a more expensive option on here (am sure others can advice). They will sort out tax aspects etc. Agree a gross salary not net.