The volume controls and the mute option are working for me within the Launcher client (or "LoL Patcher" as listed by Windows). I'm just guessing you're confused by the vast array of volume options for the vast array of different clients.
The LoL experience consists of three clients, all with independent volume settings: the Launcher client, the PVP.net client and the game client. If you hit the gear icon in the upper right before hitting launch, you will see the volume options for the Launcher client. If these are not working, in addition to adjusting the volume for "LoL Patcher" within the Windows mixer, I don't know what to tell you other than suggest that perhaps something is wrong with your sound driver. I mean the fact that Windows mixer, which is totally independent, is not affecting sound playback is, if true, completely unexpected behavior to say the least, and wouldn't be a problem with the client.