An interrogative is always either "asked" or "enquired" in reported speech. Hence the subject and verb of the reported speech would be, "I asked".

The pronoun for the listener, in reported speech, will change according to the object of the direct speech, which in this case is "her".

The interrogative sentence is beginning with an Auxiliary Verb - "Don't". Thus, the indirect speech will have to be joined by using either "if"  or "whether".

Thus, the speech will look like, "I asked her whether"

Since the principal reporting verb - "said", is in the past tense; the indirect speech will be in the past tense. Thus, it would be written as, "did not know the way home"

Hence the overall sentence will be, "I asked her whether she did not know the way home."

Thus, Option A is correct.