How can I get my mother released from a nursing home?
Her sister put her there without my permission.
That depends on a few things. If you are your mother's legal guardian, it should be pretty easy: have a copy of the order of guardianship to give the nursing home, and that should be pretty much that, because whatever your sister signed, in her own name or your mother's, or that she had your mother sign, is worthless. The nursing home can go after your sister, if they want.
Otherwise, there are too many different things to be considered, to give you any kind of simple answer. If your mother is able to live in something like assisted living, if she doesn't really need nursing home care now, that makes a big difference, for example; otherwise, the question quickly becomes, why not this nursing home, now that she's there?