I think we actually see Emma showing sympathy for Harriet's illness. I also think she empathizes with her, can understand her longing for 'home and mother' so to speak when she isn't feeling well, something Emma herself probably appreciated and felt growing up in her relationship with Miss Taylor. It is a simple comfort to turn to the someone who has nurtured you through childhood if possible and Emma does not want to deprive Harriet of what will make her feel better. I have to say that I get a real sense of 'caring' in Emma when it comes to people being ill. I think it is an innate part of her character which makes her the perfect daughter for Mr. Woodhouse. (She would have made a good nurse in a later era.) :)