Latch Circuit: Beginner
You don't need to complicate it with digital logic, just use an SCR.
When power is first applied no current flows through the LED.
After the button has been pressed, then the SCR is latched on and the LED is lit.
The only way to turn it off is to stop current from flowing through the SCR ie. remove power.



simulate this circuit &ndash; Schematic created using CircuitLab
EDIT with AND gate control



simulate this circuit