how to calibrate my multimeter temperature probe
If the type K probe is made properly you ought to get good results to +/- 0.5'C.
Like any ADC calibration, you test for errors in offset, gain, monotonicity, linearity and dead-bands and if desired the RMS integrated average squared error (IASE) for every value.
Find the value that is not affected by gain or temperature of the sensor, perhaps 0'C  then null offset error.  Knowing altitude does not affect 0'C but affects boiling point  -0.5'C/500ft  you can use ice water and boiling water to get &lt; 0.1'C error reference points.

null the error for 0'C
null the gain error for 100'C
test for monotonicity with a scope and D/A converter output that has been calibrated.  Dead-codes or hysteresis are a sign of digital to analog crosstalk on the ground path at binary transitions like xxxx1111 to xxx1000. If the apparent signal or Vref shifts with the binary state, you get some missing or &quot;dead codes&quot;.

Do you have a 12 to 16 bit ADC for adequate resolution and accuracy of 200 readings with 0.5'C increments or a carefully scaled 8 bit with 1/2 bit error in 256 deg C range or perhaps only 10 to 40'C   (?)
For linearity and accuracy, I prefer using transistors as diodes Vbe=Vce for measuring temp. over thermistors and get 0.3'C accuracy with some effort.  But now the LM75 is pretty useful.
Maybe useful info
This is all anecdotal info.
Your DMM specs are pretty loose so you can correct the gain error with a calculator.
If you want to compare a 1N4448 diode to say a PN2222A with BC shorted, try the diode test that supplies on your meter 300uA and has 1mV resolution.
Depending on diode leakage you may get between 300mV and 500mV @ 25'C then test 0'C and 100'C to compare.  With a diode tempco of -2.3 mV/'C that is almost your 0.5'C resolution requirement with a good low leakage transistor.