Where does this white dust come from?
This white dust is caused by water impurities, often like limestone, that gets vaporized by the ultrasonic thingy and blown around. You can often see it on the floor around humidifiers.
It's unlikely to be the cause of the damage though. That board seems to be a basic power supply, you can check the output if you have any voltage out. Visually it seems fine, so it's probably the ultrasonic transducer.
You can try to put some vinegar for a few days on the transducer (inside the water compartment) if it's due to limestone clug-up that would dissolve it.