Using a type K thermocouple ADC to measure a type E probe
This is how I see it: -


You have a K type TC connected to a proper measurement interface
The above is telling you 550 degC
Your TC in your soldering iron is telling you about 300degC when connected to your "proper measurement interface"
You have concluded that the TC in your iron is an E type.




Looking at the graph above you would expect an E TC to produce a bigger voltage than a K TC for the same temperature so, you have to ask yourself if your beliefs are founded.



EDIT section

It now appears that the OP is using the internal TC connected to a K type interface circuit and this does tend to justify that the likely internal device is an E type TC. The graphs are fairly linear and both fall through 0 uV and 0 degC so a first order approximation is to treat the conversion as linear. At 1000 degC a K type produces 41 mV and an E type produces about 76 mV.

If you want a more precise polynomial try THIS data sheet: -