If the nanny is self employed, then she submits you an invoice for the services provided, eg:
5 x hours nanny services @ 10.- per hour = 50.-
Exactly the same as the auto mechanic that repairs your car: he also invoices for hours worked, and you do not organise his sickness insurance.
All the issues around holidays, sick leave, insurances, etc become the nannies problem.
She only invoices you for the services provided, nothing else.
You only pay what is on the invoice.
From your previous posts it appears you are cost-sensitive, so be aware that the costs of a nanny service for you will be HIGHER, because the nanny determines the hourly rate, which includes allowances for holidays, sick leave, insurances, etc etc etc, and you cannot manage those costs.
The nanny service does however shift the paperwork burden from you to the nanny.
NOTE:
The paperwork for an employer of house staff is only hard the first time you do it. All subsequent years are easy. You just have to keep good records, and be diligent with your record keeping.
Generating a monthly pay slip is easy (fill out a PDF form)
Generating an annual Lohnausweis is also easy (fill out a PDF)
Filing the AHV stuff is a bit more complex, but once done it is easy to repeat.