모르 는지 모르겠네요 is wrong. You're basically saying "I do not know I do not know." Both 모르 는지 and 모르겠네요 mean I don't know. It is better to say 모르겠네요 or 모르겠어요.

Can you give a full sentence? I'm not sure about that but I think it means "I don't know whether one (he or she) doesn't know".. As in "I don't know whether he knows or doesn't know about my feeling toward him, oh yeah, pretty flowers, 오빤 강남 스타일 ♫♫"