Why won't my 2 x NPN circuit latch?
Currently the LED is lighting because your switch provides a direct path, it's not the transistors working. There is no means of supplying current to the bottom transistors base with the switch open (the emitter output will always be lower than the base, so it cannot provide current)  

You need something like this (M1 can be a PNP transistor if desired):



When the switch is pressed, current flows through Q2s base, so Q2 turns on, pulling M1s gate low and also turning it on. Once M1 is on, R4 now provides the base current to keep Q2 switched on when the switch is released. 

Simulation: