Signal differences into Labview to those measured before Multiplexer
This could well be due to the fact that your ADC is a Delta-Sigma and therefore looking for a varying signal and not a relatively static one from the thermocouple. I would bite the bullet and just buy the relevant DAQ device from NI. Once you have it it will take minutes to measure the signal. 
Or try getting a temperature sensor with built in converter that gives the output as I2C that means there will be little room for conversion error, as you will also have many dissimilar metal joins all adding an effect to your thermocouple signal.