Here is what I think, but you will need to do a little legwork to verify things:
The other 4 16/4 wires that you mentioned that are not currently connected to anything go to the volume controls. I'm sure they are not labeled because they never are (NONE of the wires in my house were labeled).
One goes to the input of one VC and the other goes to the output of that VC. Likewise, the 2nd set goes to the other VC in the same manner. Normally the ones connected to the VC outputs would have been run to the speakers but instead your speaker wires go straight from the speakers to the same room and currently have no relation to the VCs.
You need to figure out which wire is which and I'd start with one speaker and VC only. If you want to work on the VC to control the game room, disconnect the wires from the Niles until only the game room speakers play and mark those wires so you remember they are the game room speaker wire.
Now you need to determine which two of the other 4 16/4 wires go to the VC you want to be the game room VC. You can do it by trial and error or better yet go to Home Depot and buy the LanTracker ($40) to trace the wires. You can take the VC out of the wall (it's just like a light switch) to identify the wires using the LanTracker.
Whichever wire is connected to the output of the VC needs to be connected to the game room speaker wire. The one connected to the input needs to go to the receiver or to the Niles if you want to keep it that way.