"Miss Brill" is the story of, as you say, a "sensitive, lonely, and sad" woman. The truth is that she is all of those things from the opening lines of the story; however, her eyes aren't opened to that truth until the incident with the young couple on the bench. She's not crazy, but she does talk to her fur. She has no family and, when she ponders telling her observation about life being a play to someone, she thinks of an old man she reads to and a few students she tutors in English. She talks to no one while at the park but creates stories about their lives in her head. Once her eyes have been opened, she sees what a lonely little creature she must appear to others--which she actually is. She no longer finds joy in the simple pleasure of a nut in her bakery treat, and she cries as she returns her fur into its box--and re-enters the sad, lonely confines of her "closet."
I've attached a great e-notes link below if you need more help. Good luck with your essay.
I think that you have stumbled upon the essence of the story. It seems to me that one of the most profound elements in Mansfield's short story is the power of the threshold of revelation, to quote Tony Kushner. The threshold of revelation in the story might not be one that Miss Brill experiences, but it is one that we, as the reader, do. When the couple on the park bench derides Miss Brill in such a painfully insightful manner, we understand that she is lonely to have constructed this alternate world where she is the center of it. This revelation is a sad one because we recognize the reality of Miss Brill's existence as one that is not real. She also proves herself to be sensitive by being mindful of the criticism aimed at her. She blames the stole for it, proving her level of sensitivity about such a condition. It seems that being able to analyze her experience in the park might be the best way to fully grasp how she is sensitive, lonely, and sad.