LCD backlight connection, K pin connected only
Sometimes there is a place to mount a series backlight resistor right on the LCD display- maybe to the +5V pin 2. Probably something like 100 to 220 ohms is appropriate, sometimes as low as 50 but take care not to damage the backlight. 

There is a lot of variation between different common LCD displays for the backlight connections. 

For your newer module, you would connect A to +5 through an appropriate resistor and K to the MOSFET drain. The LCD datasheet should supply guidance as to which of 15/16 is A/K (usually 15 is A, 16 is K) and what resistor is required. Sometimes there's already a suitable resistor mounted on the display, but don't count on it or you could fry the display. 

The original circuit appears to have had a resistor on the display, as none is shown on the schematic.