。
She asked me whether I knew his address.
Literal
She [topic-は] me [to-に] his address [object-を] know-whether-or-not [object-を] asked.
~かどうか ('whether or not') is the standard grammar for forming embedded yes-no questions in Japanese. Built from か ('question') + どうか ('how'), it creates a noun phrase you can plug into any verb that takes an object — here 尋ねる ('ask'). The full embedded clause 彼の住所を知っているかどうか ('whether he knows his address') becomes the object of asking, marked with を. This kind of layered embedded question is grammatically subtle but widely used in everyday Japanese.