make sure you didnt' hold down said key when booting up your device. If it is still an issue,try to clean around the button with a little alcohol. If still no go, then you have worn out the button itself.
If nothing is turning on after the overheating, do check that nothing got accidentally unplugged on the motherboard. If there is absolutely nothing unplugged, such as fan leads or perhaps even sensor leads, then the overheating damaged a component on the motherboard. Hard to say exactly which part though...
This is one of the main problems with home espresso machines. They all literally only last a few years, five at best i've heard. The problem with yours is probably a blown seal, much like an o-ring, some where in the wand connection. If you can dis-assemble the device, voiding any remaining warranty by the way, you can try to just clean the o-ring area and it may become usable for another year at the most. Contact the vendor if possible for an OEM replacement o-ring so that it matches in size and thickness correctly.
You do not need an anti static mat or wrist strap. It is best to have it plugged up, and unplug the power connector off the motherboard. Make sure you have one hand, or your forearm touching some metal part of the case while working. I am a field tech, and have never carried around a mat or strap. Just take the other necessary precautions.
The charging port should be a standard mini-USB connection. Try hooking the phone up to your computer to see if it shows any sign of life. If so, double check that working charging adapter with a friends phone, or use their charging adapter. If it doesn't even show signs of life plugged into a computer (should try to charge itself from the computer) then the USB port on the phone is damaged, and is not so easily repairable.