When a sentence begins with not only followed by but also, and not only starts the clause, the word order after not only must be inverted like a question. This means we need an auxiliary verb, such as did, was or is, before the subject.
The correct form is: Not only did she go to Kashmir but also stayed there for a month. Here did comes before she, and the main verb went changes to its base form go.
The given part A, Not only she went to Kashmir, keeps the normal subject-verb order without the auxiliary did. This breaks the inversion rule, so this part is wrong.
Parts B and C, but also stayed there and for a month, are grammatically fine on their own and do not need any change.
Since the error lies in part A, the correct answer is option A.