Robin Brown is not the lady.
The initial assumption is that the "lady" is Robin Brown, the nurse. However, this immediately proves to be false. If we assume that Robin Brown is the lady, and her opening statement incites a response from the person with black hair, this means that Brown's hair cannot be black. It cannot be brown either, because then it would match her name. Therefore, her hair is white. This means that Doctor Black's hair is brown and Doctor White's hair is black. But Doctor White responds to the statement of the person with black hair, so they cannot be the same person. This leads to the conclusion that Robin Brown is a man. Doctor White's hair cannot be white. It also cannot be black because he or she responds to the black-haired person's statement. Therefore, Doctor White's hair is brown. Doctor White is not the lady either, because the lady's hair is not brown. This means that Doctor Black is the lady. Her hair cannot be black or brown, so that means her hair is white (platinum blonde).