I was just asking about the DSL filters because you indicated a 3-way tap from the one in the hallway, with one line running into the living room where you have the modem. I assume what you actually have is the living room extension plugged into the ADSL socket on the filter and a 2-way adapter in the voice socket which feeds the hallway telephone and upstairs extension.
So the Sky lines are fine, phew on that count. But interesting that plugging in the now defunct kitchen wire kills the phone. And by kills I mean, no dialtone, nothing, nada.
Ah! Now that makes sense. I expect that where the modular plug has been ripped off the end of the cord in the kitchen there are strands from the wires touching each other. When you plug that cord into the jack it is shorting out the line, hence no dial tone (it will also put a permanent signal condition on the line so that anyone trying to call you will just get a busy signal).
If you strip back a little of the outer sheath in the kitchen and separate the wires so that there are no shorts, you should be able to plug that extension in again without knocking out the other phones. If everything else still goes dead, then the tug the cord got has probably damaged it elsewhere so that there are more shorts further back along its length.
Either way, remember what I said above about voltages on the line, and unplug it again when you've tested, until such time as you repair it.