If a thermocouple compensating cable is broken. can i use an attach to fix it or i must install a new one?

possible but better is the new one for the accuracy of signal.

can attach, but no guarantee for accurate output as it depends on resistance.

you to weld with the same type of the material of thermocouple type material ie if it type K u need to use alumel and chromel only 

The compensation is usually an RTD sensor which is based on the resistance change according to the temperature.

So if you make a good attach that does not affect the resistance then it is accepted. 

Try to fix one junction box to joint it, but you will loose accuracy