The letter K in the professional name J.K. Rowling represents Kathleen.
Rowling was born Joanne Rowling and did not have a middle name at birth. When Bloomsbury prepared to publish Harry Potter and the Philosopher’s Stone, the publisher asked her to use two initials rather than her full first name. The decision was intended to make the book’s author seem less clearly female to potential boy readers.
Rowling selected Kathleen, the first name of her paternal grandmother, for the second initial. She therefore became J.K. Rowling in the publishing world. The name does not stand for a second birth name added later in the usual way.
This detail is often confused with the author’s use of the name Joanne Kathleen Rowling in biographies and reference works. The added “K” is a chosen professional initial, while Joanne is her birth first name.