J.K. Rowling used Kathleen as her middle name for publication when her publisher wanted two initials for the Harry Potter books.
Rowling was born Joanne Rowling and did not originally have a middle name. Bloomsbury asked her to use two initials because it expected the books might appeal to boys as well as girls. She chose K in honor of her paternal grandmother, Kathleen Ada Bulgen.
The initials became part of one of the most recognizable author names in modern publishing. Rowling has generally been known publicly as J.K. Rowling rather than by her full name, Joanne Kathleen Rowling.
The K does not stand for a name given to her at birth. This is also separate from Robert Galbraith, the pseudonym she later used for crime fiction.