An external regulator on a heatsink is the way to go. Certainly take the LED's off the 5V pin, the ethernet shield presumably plugs into the 5V pin anyway and you'll have to live with that (about 150ma??)
That switchmode power supply should be giving out 5.0V +/-0.1V if it was any good at all - looks like its cheap dangerous junk: bin it. You may have damaged the Mega (6V is the abs max for the ATmega microcontrollers).
Give the seller negative feedback if that supply really is giving out 6.6V...