Check each diode and make sure one isn't shorted or open.
Check the small capacitors on the primary side, they're part of the start-up circuitry and if they've gone bad they may not give enough power to the power supply controller IC.
I see you have voltage on the big capacitor, this means your fuses are good and the bridge rectifier is working, so voltage reaches the capacitor. From there electricity should go to the transformer
That DIP8 chip that says A6159 is the switching voltage controller :
http://www.irf.com/product-info/datasheets/data/iris-a6159.pdfThere should be 170v on 5th pin , and a lower voltage on pin 3 at least 3v and at most I think 35v but usually it's around 10-20v (pin 1 is at the bottom of the text on the left, pin 2 is the next one to right pin 5 is the right most pin at the top and so on)
Check for continuity between the drain and source/ocp pins of the chip (see datasheet for pinout), just in case the internal mosfet is shorted.
IF one of the ccfl tubes is broken or the wire got desoldered, the backlight circuit will turn off in about 1-2 seconds to protect itself. Same if one of the tubes is too old and draws too much current. But you should be able to make the monitor work without the backlight connected (just point a flashlight to the lcd display to see the image)