Type-K Thermocouple interface problems
I suspect that you are overlooking the input offset voltage of the AD8223. The data sheet says it's typically 250 uV. Multiplying by your gain of 106 gives an output offset of 26 mV, so that would more than account for your problems.

Try shorting the inputs together and look at the output. Subtract that number from your 5 mV, and see how that works.