The code and pins 1 <--> 14 have absolutely nothing to do with the backlight.
The backlight is basically an LED or more likely an array of LEDs that you can treat as a single LED.
You need the positive lead of a 5v DC supply connected through a resistor to the anode (+) side of the backlight and you need the negative lead of the 5v DC supply connected to the cathode (-) side of the backlight.
So, as long as the backlight is still good, and they are relatively hard to destroy, you just have to get the anode and cathode connections identified. The problem is that your datasheet has conflicting information. The 'Mechanical Diagram' shows the end pin as (+) but the 'Description of Terminals' on the next page implies that the end pin may be (-).
You have wired your circuit in accordance with the 'Mechanical Diagram' so I would now try it the other way. Move the blue wire from pin 15 to the (+) line and move the purple wire (attached to the resistor) to the (-) line. This puts the resistor in the cathode lead instead of the anode but that's ok.
Don