You can now inspect the thermostat. This comprises a bimetallic strip which opens or closes a pair of contacts. The contacts may need cleaning as a result of ingress of water.

You can clean the contacts with switch cleaner fluid (available in an aerosol) and a small piece of fine emery paper. Ensure that the contacts snap open and closed when you apply light pressure to one side or the other of the bimetallic strip.

On this iron, perhaps slightly unusually, there is a second pair of normally closed contacts which may have suffered corrosion. If you know what these are for, please leave a comment on this Instructable. My guess is that they are a safety cut-out to prevent overheating and fire if the iron is left standing flat while switched on.

Identify the connections to the element and test the resistance between them with a multimeter. It should be around 30 ohms (this iron is rated at 240V 2KW).