A few things:
K is a common thermocouple
let's say if you wanted room temp or cryogenics then use T Type

Wire is made with limits of error. It's not great for knowing the traceable temperature. +-1 degree is common.

Probes are limited to your application. Some Kapton surface probe would be limited by the Kapton

What are you going to be measuring? Pipe clamp?

Readouts have another problem. I've had some that would start measuring at 250 degrees Centigrade. Not likely for your type of device.

Quick meter test. Short the inputs with a piece of copper wire, If the device is capable of measuring room temperature, it will read room temperature.

There is thermocouple wire and thermocouple extension wire. Don't confuse the two. You don't make junctions with extension wire.

Multiple parallel junctions average, Even when they are at the tip.

They are color coded. Yellow outer jacket is K. The inner conductors are colored too. RED is always negative for thermocouples in general, (at least in my experience). I've used R,S, J,K, T and C.

I'm familiar with mineature and full size connectors.

The junctions (or two 1/2 junctions) should be isothermal.

Old school was two thermocuples in series and one in an ice bath. Then use the tables (reference 0 C) to convert voltage to temperature.

New school. Use a precision semiconductor sensor an measure the temperature of the terminals. Do a reverse lookup and use that mV value. Add or subtract, then look value in a table or polynomial.

Thus the ability to ambient compensate is limited. The temperature where the meter is located.