Translation:She does not know anything about your family.
No, "sa" doesn't change whether the subject is lui/lei/Lei (he/she/you (formal))
Since there is no subject pronoun this sentence could be translated as "He/She does not know anything about your family" .
Additionally I don't think it could be translated as "You do not know anything about your family. " because for that to work the Italian sentence should be: "Non sa niente della Sua famiglia"