どこかへお花見に行きませんこと。means どこかへお花見に行きませんか: Would you like to go to see the cherry blossoms somewhere? >どこかへお花見に行きましょう。Let's go to cherry blossom viewing somewhere.
行きませんこと: This ことis used in lady's spoken Words at the end of the sentence to be elegant. These lady's words are called, お嬢様言葉の「ことのて言葉」、which is 「~こと」, 「~の」、「~て」are used in the end of the sentence in lady's speech. It is ultimate humble language.