others/others'

Both are correct but have slightly different meanings. In the first sentence, others is functioning as a noun, so it refers to other rooms that she likes to clean. In the second sentence, others' is functioning as a possessive and so refers to other people's rooms that she likes to clean. In the first sentence, the other rooms don't necessarily belong to anyone; they could be living rooms or kitchens. In the second sentence, the other rooms belong to people, which means they are probably bedrooms.

Thank you very much, Shapeless.