In 1919, he married Lily Mary Batterham, with whom he had three daughters: the late feminist, Marxist anthropologist Eleanor Leacock (1922–1987); musician (Jeanne) Elspeth Chapin Hart (born 1920); and writer and poet France Burke (born c. 1925). He later divorced Lily and, in 1933, married her sister Elizabeth Batterham with whom he had two sons, Michael and Anthony. Burke served as the editor of the modernist literary magazine "The Dial" in 1923, and as its music critic from 1927 to 1929. Kenneth himself was an avid player of the piano. He received the Dial Award in 1928 for distinguished service to American literature. He was the music critic of "The Nation" from 1934 to 1936, and was awarded a Guggenheim Fellowship in 1935.