The first challenge with modern steam irons is often how to disassemble them, and unfortunately the repair options are often limited, but some of what can be attempted is described here.
Steam irons are quite often seen at Restart parties. They can be challenging to open, and as with any appliance which mixes water and electricity the fault can often be due to a failure, ultimately, to keep them sufficiently apart.
Usually there will be one or two easily accessible screws at the back of the handle but these may only release that part of the handle. Another screw or screws near the front may be hidden for example under the steam button, but even after removing these the handle cover can be tricky to remove. Often there will be a series of clips down the sides, but this may not be all. In the iron illustrated there was another clip to be released on the top just forward of the cable which required the whole of the top cover to be pushed forward.
With the top cover off it should be possible to find the screws which release the whole of the plastic top, in order to access the thermostat etc. and (if not already accessible) to replace the mains lead.
On reassembly, ensure that any tubes associated with the water reservoir and steam delivery are properly connected.
Because irons are moved around a lot, bending the mains flex, one or more of the conductors can break, usually close to the iron body. Locating the break may be tricky, although usually it will be close to the body & can often be felt by bending the flex conductors individually.
Either cut the current flex back to before the break then reconnect it or replace it with a compatible heat-resistant flex.
The thermostat switches the element off when the iron gets up to temperature and back on when it cools down a bit. Usually it consists of a pair of contacts, one of which is a bimetallic strip which bends away from the other when it heats up. The distance between the contacts is adjusted by a threaded shaft attached to the temperature control knob.
If the electrical continuity is otherwise good and the element shows the correct resistance, but the iron won't heat up, try cleaning the thermostat contacts and checking that they actually do contact electrically when they come together.
If the iron is heating up but isn't producing steam, this is almost certainly a mechanical problem, usually caused by limescale: descale it.