I am having exactly the same issue mentioned here. My Latitude E6420 shuts down after 5 seconds. While starting only caps lock key goes on for few seconds. This happened a week ago, but then I tapped hinges as per verified answer in above link and it started but now again it is causing the same issue.This time tapping of hinge didn't worked. So, I disassembled it and after reassembling it started once and again it stopped. I am sure that LCD is not broken. But can't figure out the actual reason for it.
Here is the Service Manual for the out of warranty 2011 Latitude E6420.
* Disconnect the AC adapter from the laptop
* Remove the battery, page 21
* Press and hold the power button for 30 seconds
* Reconnect only the AC adapter to the laptop
* Press the power button. What happens? What are the LED codes?
Unplug the system, remove the battery and hold the power button for 30 sec. Verify that the LED on the AC adapter is ON before you plug in the notebook end. If it's not, replace the adapter.
Try powering up on AC only - if it will, but won't with the battery connected, the battery is bad.
If the system still won't power up, the mainboard is bad.