Flat characters are relatively uncomplicated and do not significantly change throughout their story. Round characters are more complex and they do develop, sometimes in surprising ways.
Miss Brill does develop a bit during her story, though it's difficult to be certain that she changes enough to qualify as a round character. At the beginning of the story, she doesn't seem to realize that she is just like the other people at the park who are "odd, silent, nearly all old," and they seem to be unaware that they look "as though they['ve] come from dark little rooms or even -- even cupboards." Miss Brill doesn't realize that her silent eavesdropping makes her an oddity and that she too lives in a "little dark room -- [...] like a cupboard." By the end of the story, it does seem that she's developed some awareness of her condition as a result of hearing the young couple's unkind conversation about her. Miss Brill skips her cake on the way home and is evidently crying when she puts away her fox fur. However, she doesn't seem to understand that it is her own crying she hears: the narrator says that "when she put the lid on [the fur's box] she thought she heard something crying." We understand that Miss Brill hears herself, but she does not recognize it, and so she seems not to have developed a complete understanding or sense of self-awareness.
Whether Miss Brill is a complex character is similarly open to interpretation. Miss Brill is an older woman with a well-developed imagination that compensates for her lack of friends. She does not realize how socially obsolete she is until she overhears a young couple talking rudely about her; at this point, Miss Brill begins to develop some awareness of the way others see her, but her imagination still protects her, to an point, from recognizing the full extent of her alienation. Though her struggle may be representative of something many experience (how often have we disparaged a quality in someone else only to discover that very quality in ourselves?), Miss Brill experiences a complex range of feelings as she struggles with both pain and denial.
Miss Brill is a round character because a round character is a major character in the story that is influenced by conflict and is actually changed by it.
Throughout the story, the tone of the music mirrors Miss Brill's feelings. To begin she is a people watcher who loves sitting in the park listening to her "music". This day Mansfield narrates:
There were a number of people out this afternoon, far more than last Sunday. And the band sounded louder and gayer.
As the story finishes, she is consumed by a couple who talk poorly about her right in front of her.
"But why? Because of that stupid old thing at the end there?" asked the boy. "Why does she come here at all–who wants her? Why doesn't she keep her silly old mug at home?"
This bursts her spirit. She had thought she was above everyone else and a great observer of all other characters, but alas, here she is judged. It forever changed her attitude from one of happiness to complete melancholy.
But when she put the lid on she thought she heard something crying.