Your right, not much can go wrong and take down both items...
Either
1.) The power is being lost in the cord, or shortly after it comes into the appliance. That one BLK connector where 3 wires meets looks suspicious. Take your meter, set to AC voltage , place black lead on ground and place red lead at that connector, OR at either of the switches L side at the bottom or top. Should read 110. WAIT WAIT WAIT... If you know how to safely use a meter you already know this... so im assuming you dont know how, at which point I tell you not to attempt it... And if you do im not responsible... etc etc...
2.) The neutral path is open...
I suspect Its one of those 2. They just dont share enough in common for both of them not to work.