。
She asked me whether I knew Tom's address.
Literal
She [topic-は] me [to-に] Tom's address [object-を] know-is [whether or not-かどうか] asked.
~かどうか is the standard frame for 'whether or not X' — a yes/no question packaged as an embedded clause. It combines the question particle か with どうか ('how-or-what'), forming a compact 'is it X or not.' The matrix verb 尋ねる takes this whole bundle as its content. Note 知っている — the present-progressive of 知る — which expresses the resulting state 'know,' not the moment of getting to know. Japanese marks this distinction systematically: 知る is the becoming-aware moment, 知っている is the resulting state of knowledge.