Which NTE5 faceplate is it? Is it the simple NTE5A with just the single phone socket at the front & they need to use a dangly filter to connect the router?
If it is a more complicated 1 it might be safer to go back to the simple 1.
There must be an extension socket somewhere in the house as you have wires coming out the back of the faceplate. You need to track it down. It could well be causing the issue that the QLT is noisy with the faceplate on but quiet when connected straight to internal test socket.
does that leave the incoming wiring (White/blue, Orange/White, White/Blue) as the issue? Where does that wiring originate and does it constitute the owner's property (and thus her problem)
Yes, that is the issue! Where indeed? It is owner's property & problem as it is outside BT's test socket.