I started by only connecting the backlight (pin 1 = backlight anode, pin 2 = backlight cathode). Pin 1 was connected with a 220 ohm resistor to 5V and Pin 2 connected to GND.
As you may have already noticed the LCD displays that have their pins in the lower left corner have a non-standard pin layout so you have to be particularly careful. Your description of the pin connections is correct for your display but the 220 ohm resistance may be too high since the calculated value for a white backlight is around 80 ohms.
What bothers me is the statement that "The backlight turned on and then turned off after a few seconds". One possible explanation is that the series resistance was not connected properly and that excessive current destroyed the LED backlight after a few seconds. On the other hand, from what I have observed from other forum posts is that these LEDs usually survive this kind of mistreatment, possibly because the power supply shuts down. Are you sure that your power supply is still functioning?