Using old laptop LCD as external VGA screen
At least the wrong controller...

You've got a lot of problems with the hardware I can see, but at a minimum:


Your display is "HD+" which is not supported by your controller. It's barely within the practical capabilities of the analog VGA port you're trying to use.
The panel is 3.3V, not 5V, so setting it to 5V may have fried the panel's onboard LVDS controller.
The panel from Sharp uses the FFC interface, but the cable in your photo is a pin header type. That means either, you have the wrong cable, or this panel is an OEM-specific version of the standard Sharp panel (quite possible since you removed it from a high volume OEM product). The pin-out of the cable may be non-standard or different from your controller board -- in which case you've either damaged the I/O electronics in the controller chips, or forced the controllers into reset (shutdown).


Some other notes


The backlight flash you see at startup is normal and expected. The panel is all white (all pixels open at startup) and then all black (all pixels closed) while it waits for the controller (IIRC for this family of Sharp panels). The backlight is still on, but blocked by the LCD after it's startup procedure. Or... the backlight controller could go into standby after startup by not detecting control signals from the controller (as it's already entered reset -- see above).
Post the part number from the controller IC in the middle of your controller board if you want to confirm it's the wrong controller board (but I'm pretty certain it is). Also, post a photo of the other end of the "LVDS" cable (where it's connected to the panel).