'''HER Zygiel (Herzygiel)''' This just for an FYI. As you already know, your computer and your LCD are not communicating. Hence the LED 4 off. No, you cannot test the LCD without having special equipment at least an LED backlight tester would be required. If you have the equipment you of course could, but it is unlikely since you are asking.
Diagnostic LEDs (A)
Indicates that the trickle voltage from the power supply has been detected by the main logic board. This LED will turn on when you connect the iMac to a working AC power source. The LED will remain on as long as the computer is on or asleep.
When the computer has been shut down correctly, LED 1 behavior may differ:
If a startup event is scheduled in System Preferences/Energy Saver, LED 1 will stay on after a correct shutdown.If no startup event is scheduled in System Preferences/Energy Saver, LED 1 will turn off and will stay off as long as the power cord is kept connected and an AC power source is present. Disconnecting the power cord and plugging it back in will turn this LED back on, even if the computer is still off. After disconnecting and reconnecting the AC power source, this LED could remain off:
If the AC power source is missing or disconnected. If the logic board is disconnected from the power supply or the AC receptacle. If the power supply board is faulty.
Indicates that the computer is turned on. This LED will be on as long as the computer is turned on (but is not asleep)and the power supply and voltage regulators are working correctly.
Indicates that the computer and video card are communicating. This LED will be on when the computer is communicating properly with the video card. If LEDs 1 and 2 are on and you heard the startup sound but LED 3 is off, the backup battery (on back of logic board) may need to be reseated or the video card might be installed incorrectly or need replacement.
Indicates that the computer and LCD panel are communicating. This LED will be ON when the computer is turned on and a video signal is being generated. If LED 4 is on and there is no image on the display, the LCD panel or the cables between the LCD and logic board might be installed incorrectly or need replacement.
The part that shorted out for whatever reason is reference designator L4400 which is a 120 Ohm 3A fixed inductor (ferrite Bead) in a 0603 package. A Murata BLM18KG121TN1D would be a perfect fit. They are available at Mouser.com Digikey.com Farnell.com and of course ;-) on ebay https://www.ebay.com/itm/Fair-Rite-0805-...
Without the component you are not getting the 12V to the LCD. No power to the LCD - no 4th LED!