According to this site (which sounds right to me) the two inputs are solid orange and solid white, which is consistent with Mark's photo. The orange wire should connect to the broken-off drop-wire.
Note the site says the drop-wire contains steel cores for strength to be left alone.
Frances explains how the internet is still working – it's carried by a radio signal that can jump gaps.
Once repaired, the phone should work normally again but your broadband may stay stuck at the slower speed. It's because the system adapts to a poor line by dropping the speed. Once it's decided a speed for the line an engineer might have to reset it to get back to normal.
I had to jump through call-centre hoops before talking to the right person; I had to persist that the link speed had suddenly dropped. The ordinary call-centre script doesn't seem to cover that – it's tuned to human error, and – although they test the line, they're happy if it works at all. After explaining it was a severe line speed drop, maybe due to a thunderstorm, I got past front line support and talked to an expert who understood the problem. A BT engineer reset the link at the exchange and then came to the house to test it. Worked better than before.
Dave