Yes thermocouples can be extended, but you have to use a 'compensating' cable, i.e. typically multicore cable made from the same metal alloy as the thermocouple itself, but normally not so pure and therefore cheaper.
You also have to connect via "compensation" plugs and sockets, again made from the same material.
If you need 30 meters extension, you will probably have to check that the resistance of the extension cable does not exceeds the maximum resistance that the converter can operate with.
Also a shielded cable may be needed to reduce noise from other electronics and the main line.
If the thermocouple sensor is made of 'simple' K wire, it can just be cut and shorted anywhere to form a new thermo junction at the short.
May be it's a good idea to isolate the junction with epoxy if the temperature to be measured do not exceed 100 deg.C.
Normally a thermo junction is 'spot' welded, but it often works with just tighten the wires hard together or soldering them together.
If the broken thermocouple is inside a thin metal tube, it can usually not be repaired.
Here is an introduction to thermocouple and theory:http://www.omega.com/temperature/z/pdf/z021-032.pdf
And some more:http://en.wikipedia.org/wiki/Thermocouple
The word "thermocouple" gives over 9 million hits on Google, so it's not hard to find theory and practical hints.