The "black box" as you call it is a standard microswitch, readily available on the internet. On the other side it should have markings for the voltage and current ratings, so just find one that has the same or greater ratings on one or both of those values. Do a search for "micro switch normally open" and you should come up with dozens of options. They're all a standard size, so as long as it has the same connectors as yours and the ratings are as good or better, any of them will work.
If you have an ohmmeter or continuity tester you can check to see if it's still working; just unplug the two orange connectors, place your probes on those two tabs and press the little button on the lower left side as shown in your picture. Since yours is a "Normally Open", or NO switch, there should be no continuity when you're not pressing the button, and it should show very little resistance when you press the button.
Replacing it should be simple, just unplug both wires and unscrew that plastic piece in the lower right corner and transfer it to the new switch.
With any luck, that could be all that's wrong. You could test out that theory by bypassing the switch with a jumper wire, but since it will probably be carrying full household current, either 110 or 220VAC, use a decent size of wire and definitely make sure you're not touching it. Note that it is probably a safety interlock to ensure the lid is on before it's energized, so be aware that it could come on when you're not expecting it once you plug it into the wall socket.
Good luck; let us know what you find.