Miss Mills is a friend of Dora's. She is a couple of years older – around 21 when David and Dora first meet. She has been unlucky in love, so she looks at David and Dora's romance from the remote position of a girl who has completely given up on love. David finds her pose of extreme experience and old age kind of hilarious, since she is, after all, only 21.
Dora's friend agrees to carry messages between David and Dora while they are courting. She also keeps a diary of Dora's movements that she passes on to David to comfort him while he cannot see her openly (since she's the boss's daughter). Miss Mills goes to India with her father before David successfully marries Dora, leaving David without a go-between when Dora goes to live with her aunts.
There's an odd epilogue for this background character. Miss Mills is mentioned in Chapter 64 – the last chapter – having married a wealthy Scotsman and settled down into high society life. David describes her as spoiled and hard to please. This little jab at Miss Mills provides a broader critique of wealth, which has ruined her generous good nature.