First you need to obtain an entry permit / employment visa so she can enter the UAE. Then you need to apply for a UAE residence visa for the maid. The DNRD will issue a residence visa, labour card, and labour contract for the maid....
By the way...don't be surprised if a nanny does not want to sign an "official" nanny contract. Some are worried because they may not be legally allowed to work in the U.S. and as such they don't want to sign something that could come back to hurt them. Some (and this may be...