VisitKazakhstan is a state run website which provides a wonderful resource for learning about the different instruments and kinds of ethnic music in Kazakhstan. With sections covering instruments including the dombra, kobyz, sybyzgy, astayak, sazsyrany, zhetgen, shankobyz, daulpaz; vocal artists including Jayau Musa Bayzhanuly, Birzhamn-sal Kozhagululy, and Akan-seri Kormsauly.
From Folklore to Soviet National Culture - The Process of Formation of "Kazakh National Music" (1920-1942) is a nice brief essay provided by the Slavic Research Center of Hokkaido University which follows the history of the development of national music in Kazakhstan.
"Музыкальное наследие Казахстана | Musical Heritage of Kazakhstan" was a neat little website with written history of Kazakh musical culture, composers, performers, researchers, young talents, ustruments, legends, and musical works. Luckily, the website has been preserved by the internet archive, and is still acessible in its original form.
New York Public Radio Archive offers an hour long playlist of "Contemporary Music from Soviet Central Asia and Kazakhstan.," possibly recorded in 1965.
Music of Central Asia has a chapter on Traditional Kazakh music with video and audio recording of different traditional Kazakh instruments.
Kazakh.TV has an entire programme dedicated to legends of music in Kazakhstan. Check out their website for clips from the show.
The musical heritage of Kazakhstan was to be carried on to new generations through a totally noncommercial and humanitarian project called Asyl Mura, the most comprehensive music portal about Kazakh music and its history. Unfortunately the website is no longer running, however it is still browsable through the internet archive.
Asyl Mura - Musical Heritage of Kazakhstan intended to restore within five years roughly about 3 thousand hours of pure sounding music.
The project would enable the current and future generations of Kazakhstan listen to their musical heritage, thus understand and appreciate the rich and unique culture of their ancestors. The project accomplished this goal by transferring the wax tapes, which were not compatible with the current music systems, to modern digital recordings and save them from inevitably getting lost. Asyl Mura program includes prospecting work, systematization, restoration and digital refinement, recording on modern audio carriers of the best folk, classical and modern music of Kazakhstan