Expat, they must be "importers". People come here exceptionally for work. No one is coming here because he/she got a "green card" or "Landed Immigrant status", like in US or Canada, and after that start looking for Job. How one can expect to find "freelance" full time nanny? Even the agency wouldn't help you in the way you see it, as again the agency would have to "import" a nanny from other country and bring her in Qatar under your sponsorship. Once this nanny is done working for you, she has to be sent back home. This is a Qatar law. Therefore no free nannies are available.
James3, you have to apply in the Immigration/Labour Deptmt's for the permission to bring your own nanny.
Good luck.