Assuming your meter is set correctly and you are making good contact, then it is likely you have an open thermocouple. But when you measure at the long length of wire, you are picking up induced noise. The other explanation for the 5mV reading of the lead wire is that you have a shorted lead wire somewhere along the length (thereby producing a 2nd thermocouple).
This would be a tricky situation for troubleshooting as your PLC would think nothing is wrong and that the temperature is simply not changing.