Sick of Shadows by M.C. Beaton
It would seem that even a pretend engagement cannot keep Lady Rose Summer from trouble. Bored with endless parties, teas and balls, she befriends Dolly Tremaine, a beautiful young girl newly arrived from the country and overwhelmed by the demands of the Season. Their friendship is cut tragically short, however, when Dolly is found dead in the river, with flowers in her hair and blood on her breast.
Rose immediately summons her fiance, investigator Captain Harry Cathcart, to help solve the mystery of Dolly's death - and to keep her from becoming the murderer's next victim.