...Be played in-game somehow?

Could be nice to use those great log in musics for other purposes too!
Like, could it be music in the game main menu/Lobby? Would be even more inspiring to make changes on my runes and/or masteries!
And I think that the music for summoner's rift is fine, but wouldn't be great like a preselected list put on random ?

Just a thought to re-use those hours of work.

All the in game music can be changed manually, if you search about how to change the login music on youtube, you'll find the location of most of the audio files for the game, if you copy the music you want to play in there and replace the old music, giving the new file the same name as the old one, you'll be able to change it =]

or you could just mute the game client's sound, download the music from the composer's web site and listen to it while you play http://www.christianlinke.com/