J.K. Rowling used the initial K in her published name because her publisher requested a gender-neutral author name, and the K stood for Kathleen.
Rowling was born Joanne Rowling, but she did not originally have a middle name. Before the first Harry Potter book was published, Bloomsbury asked her to use initials rather than her first name, anticipating that boys might be less willing to read a book by a woman author. She chose K in honor of her paternal grandmother, Kathleen Ada Bulgen.
The name became internationally recognizable as J.K. Rowling. The initials do not stand for a secret pen name or a second first name invented for the books. Kathleen was added as a publishing name rather than being part of Rowling’s original birth name.
Rowling has also published under other names, including Robert Galbraith for crime novels. That later pseudonym is separate from the K in J.K. Rowling and reflects a different publishing identity.