Neither of my two meters will accept at themocouple. Bingo: excuse for a new DMM.
Nope!
No thermocouple measures soldering iron temperatures correctly!
You need the same device that Dave used! It is basically a special thermocouple that is designed to get soldered and in getting soldered, is able to measure solder temperature.
If you solder a normal K type thermocouple, you will ruin it. You will have mixed another metal with the 2 original metals, destroying the voltage generating properties of it.
Do not let this stop you get another dmm
Use this as an excuse to get a soldering iron thermometer/calibrator instead