There's one good book on the larger subject of music and Tolkien--a bit late to track it down, but maybe you can find it--Middle-earth Minstrel: Essays on Music in Tolkien, edited by Brad Eden (2010) from McFarland.Not surprisingly, Google Books has the bits online that discuss out-of-copyright poetry rather than copyrighted music.
Since the films, copyright enforcement on Tolkien's lyrics has been strict--the groups performing Marion Zimmer Bradley's settings have been shut down, and Broceliande's recording "The Starlit Jewel" was proscribed. Nevertheless, these settings can still be heard in filk circles.