Other than through music are there any recommendations in Tanach, the gemarah or other commentators which discuss the best ways, meaning most likely to succeed, to treat depression?
In general, Judaism is in favor of good health and common sense, so the Jewish response would be "go to a specialist," similar to the way Judaism treats diabetes. That being said, there are a few different sources in the broader Jewish canon that talk about depression and low moods. Most notably, Rabbi Nachman of Breslov wrote a book called Meshivas Nafesh that deals with this subject.
rabbi akiva tatz says in his book Jewish Teenager's Guide, that depression comes when one is not growing spiritually. when the soul feels life passing by and it is not doing what it is supposed to here, depression comes.
that said, one can treat (non-medical) depression by fixing times for torah study/classes, doing acts of kindness, prayer, mitzvot etc. of things which advance a person spiritually.
the talmud (Ketuvoth 59b) also says that idleness leads to depression, so putting one's head deep into torah study,etc or work would help in this regard.