Test this using the pattern rule: whenever not only opens a sentence or clause, English grammar demands inversion, just like starting a sentence with never or rarely. Compare: Never has she seen such a place, instead of Never she has seen.
Apply the same logic here. The sentence should read Not only did she go to Kashmir but also stayed there for a month, with did placed right after not only and before she.
Checking the given sentence, Not only she went to Kashmir fails this inversion, since she comes right after not only without any auxiliary verb. This makes part A incorrect.
The remaining parts read smoothly and match standard English usage, so they are not the source of error.
Hence, part A is the faulty part, and the answer is option A.