dell latitude e5540 solid power light, no other lights on, nothing on LCD

I have a Latitude e5540 that doesn't turn on (at least not easily). But when it does turn on, it can stay on for weeks/months as long as it stays in sleep mode. This has happened three times now. If I shut down or reboot the computer, it doesn't turn on again -- or at least it's very difficult to get it on and it seems to randomly correct itself and turn back on.

Here are the symptoms when it doesn't boot up:

• Screen is blank.
• The Power light on the power button and edge of laptop are both solid white, no other lights on. Power light remains on as long as it's getting power from the battery or power cable. If I press the power button to turn it off, it sounds like it's going through a shutdown process of some sort -- not like windows shutting down, but it doesn't immediately turn off. The fans will spin up for a few seconds, then it will power off after another few seconds.
• When power button is pressed, the fans turn on momentarily then spin down after a second. But they will randomly turn back on for a few seconds here and there as long as the power remains on.
• HDD sounds like it's on. HDD light on the edge of the laptop is off.

Troubleshooting so far with no noticeable difference:

• Removed the battery and power cable, held power button to drain residual power and tried to connect just the power cable and turn it on. Didn't work.
• Connected to an external monitor via HDMI to TV -- once the laptop powers on, the TV detects a new source, but nothing visible on the TV. I also tried pressing the FN+F8 (screen mode button) to switch through the modes -- didn't work.
• Removed and reseated the RAM. I also tried inserting another stick of RAM (although I'm not sure if that RAM was working) and it showed a different set of lights. Solid Wifi, Solid HDD, Blinking Battery light. From what I could tell that means the RAM isn't good. When I switched back to the original RAM, it went back to just a Solid Power light and no other lights.
• Removed the HDD, connected it to a different computer and ran tests on it using HD Tune. Nothing wrong with it.
• The last time it booted into Windows, I checked the event logs and I didn't see any errors that would indicate a hardware failure.
• Disconnected the CMOS battery and reconnected it.
• Pressed -- showed me a screen test. The screen turned various different colours then the computer turned off after less than a minute of cycling through colours
• .Pressed  -- didn't show anything on screen, but the diagnostic lights were different. This time, the Power light was solid, Battery light blinking and everything else off. According to the LED Diagnostic page it means "Memory failed to initialize or memory is unsupported."
• I tried NINE (9) RAM chips in case the original RAM was bad. Pressing Fn+Power for each RAM module resulted in a different diagnostic light combination than the original RAM module that came with the laptop: Solid HDD light, Solid Wifi light and blinking Battery which means "The memory modules are detected but has encountered an error."
• I tried downloading the latest BIOS firmware on a bootable USB drive but it didn't seem to do anything. I tried powering on the computer, but since it's not responding and nothing on screen, I can't tell if it's doing anything.

I'm running out of ideas. I found diagnostic light information somewhere but it doesn't say anything about just a solid white power light with nothing on screen.

Any troubleshooting tips?