You really aren't providing any information beyond Type K thermocouple. What type of Type K (if that makes sense) as in open junction or closed? Sheath or non-sheath? The wire gauge of the TC wire? How are you sampling the mV outfrom the junction? What is the sample rate? If using A/D conversion, how many bits? What range of temperatures using type K are you measuring?
You really need to define the application is some detail and the thermocouple in some detail to get a good answer to your question.
<EDIT>
As to thermocouple response time(s):
You may want to
give this a read. It covers a few of the considerations when selecting and using thermocouples.
</EDIT>
Ron