The box in the picture is likely where the external wire (black) finishes and joins to the internal wiring.
Mine comes in to one of those and I can usually sync at about 80/20 without problems. That bit's probably not going to need replacing.
One of the sockets you have inside will be an old style master socket. You'd be far better off with the newer master socket as it has improved broadband performance.
If there is an internal wiring fault and you haven't got a new style master socket which provides a demarcation between BT wiring and your wiring, my understand is that BT will rectify that at no cost.
You might like to ask the ISP you're taking a broadband service from whether they can arrange for a new master socket to be fitted at no cost. If there was an internal wiring issue, the BT engineer would sort it out at the same time.