I decided to bite the bullet and separate the LCD module from the LCD/keypad shield I have. Unsoldering 16 pins simultaneously is a challenge but it can be done if most of the solder is sucked out of the holes and then you wiggle the two boards relative to each other and also wiggle each pin in each hole for a while to break loose any solder that does remain. I found an SOT-23-3 transistor labeled AM1 (like this one http://www.digikey.com/product-detail/en/SMMBT3904LT1G/SMMBT3904LT1GOSTR-ND/3062754
) installed on the lower board. I got this 2N7002 FET as a replacement for the transistor: http://www.digikey.com/product-detail/en/2N7002,215/568-1369-1-ND/763366
. Soldering something that is as tiny as one of these is also a challenge and you need to have a good soldering iron with a needle tip to attempt it. After connecting everything back together it works.
I found that the following code could be used to turn the backlight on or off or any value in between.
pinMode ( BACKLIGHT_PIN, OUTPUT );
analogWrite ( BACKLIGHT_PIN, 64); // a value of 000 (off) to 255 (on) controls PWM%
//digitalWrite ( BACKLIGHT_PIN, HIGH ); // turn on backlight
//digitalWrite ( BACKLIGHT_PIN, LOW ); // turn off backlight
lcd.begin(16, 2); // start the lcd object