If you have your 5.1 already hooked up (HDMI) then you can do a few things to make this work.
What I would do if you want mirroring only when the room is on:
Connect the analog audio outputs from your source(s) to the A/V receiver on the matching inputs. (red/white RCA connections)
Page 7 of your manual shows the back of the unit and all the red/white RCA audio inputs.
http://resources.jvc.com/Resources/00/00/90/LVT1572-001A.pdf
Then, you can use the 'VCR RECORD OUT' or the 'DVD OUT' RCA connections to a amplifier/volume control setup.
You definitely will need another amplifier to run this as the JVC is really not designed, at all, for what you are trying to achieve...
You could pick up a cheap stereo receiver from Craigslist or something to do this if you wanted to. I sold my old Kenwood A/V receiver for about 20 bucks on Craigslist and it would be ideal for what you are trying to do. Kind of a 'big' product, but cheap, and will do what you want and has volume control built in. Otherwise, you will need a separate amplifier of some sort (lots out there) and a volume control of some sort. Once again, lots of choices for both amps and volume controls, depending on what you want to do.
Since you only have stereo in the second area, you should be using the stereo analog RCA connections to feed that second area. Almost all consumer A/V devices output digital surround and analog stereo at the same time.