Well, I know I left out your favorite composer whose name starts with a J. Was he Jomelli? (I doubt it.) Janequin? (A remote possibility.) Oh, well. I can't please everybody. If I can't think of something cute to say (or at least, find a good picture and/or think of a couple of pieces by them), I just don't have time to include them. So, as the saying goes, WTS!
The letter K is a bit more populous. For instance:
Kabalevsky (Dimitri) was a relatively conservative Soviet composer who, unlike some of his contemporaries, managed to stay out of trouble with the authorities. This has earned him a reputation as back-stabbing career opportunist, political coward, and artistic mediocrity. Certainly his work isn't inspired by quite the same level of genius as that of, say, Prokofiev. Nevertheless, he wrote well-crafted music that is interesting to hear, including the popular overture to his opera Colas Breugnon, a light-classical suite called The Comedians, concertos for violin, piano (4), and cello (2), four symphonies, some songs based on Shakespeare, and a number of piano pieces that are almost a "must" for young piano students.
Karg-Elert (Sigfrid) was an early-20th-century composer now best remembered for some chorale (hymn-tune-based) preludes and improvisations which are still used today, and a widely-used book of technical exercises ("Caprices") for flute. My predecessor on the organ bench that I now occupy left behind some Karg-Elert books; having heard some of his pieces played, I may have to look into these books.
Khachaturian (Aram) was an Armenian-Soviet composer whose work is, perhaps unfortunately, represented in the "standard repertoire" mainly by the Sabre Dance from his ballet Gayane. I have heard suites from both this ballet and another called Spartacus and, although some of the tunes are catchy, I have never been very impressed. He also wrote three symphonies, as well as concertos for violin, cello, and piano. Though he was a sincere Communist, Khachaturian was singled out (along with Prokofiev and Shostakovich) for party condemnation by the historic Zhdanov decree of 1948, a humiliating and terrifying experience for all three men.
Korngold (Erich Wolfgang) was a 20th century "Romantic" composer who, because of his musical conservatism, was popular with the public but dismissed by critics. As a result his "serious" music (symphony, operas, concertos, etc.) has not been heard much, though that may be changing. His film music, on the other hand, includes such classics as Captain Blood, Anthony Adverse, The Adventures of Robin Hood, The Sea Wolf, and Of Human Bondage. One of his operas, Die Tote Stadt, is said to be brilliant but virtually impossible to sing.
Krenek (Ernst) was an Austrian composer who fled the Nazis and became an American citizen. His music borrows from styles of all different periods of music, from the polyphony of Ockeghem to the atonality of Schoenberg, and even jazz - most notably in his opera Johnny spielt auf. Though his works use experimental techniques, they include pieces in many classical forms, such as symphonies, concertos, sonatas, and quartets.
Kuhlau (Friedrich) was a one-eyed, German-Danish, Classical-Romantic composer. (You thought I was going to say "purple people eater," didn't you?) A friend of Beethoven who wrote operas, a string quartet, and a piano concerto, he is mainly recalled because of his piano and flute music.
Kurtág (György) is a modern Hungarian composer whose music was introduced to me in a live concert by the St. Louis Symphony. I believe the piece was called Stele, and it was a musical lament on the death of a fellow composer. Influenced by both classical and folk music, Kurtág is fluent in seven languages! Whether his musical language is also fluent, is a matter I have yet to study in depth. His pieces have interesting titles, such as Splinters for Piano and The Little Predicament for piccolo, trombone and guitar. A lot of them seem to be in homage to or in memory of somebody or other; an interesting niche for a composer!