Unplug the system and disconnect the battery from the mainboard.  Then hold the power button for 30 sec.  Remove and reinstall the memory module(s) and the storage drive. Verify that the LED on the DC side of the adapter is ON before you plug in the notebook end.  If it is not, replace it. If it is, connect it to the system with the battery disconnected.  If the system powers up, you're set.  If not, replace the system board.  
I bought 2, out of warranty Dell Latitude 5500 laptops from a company that had gone out of business for my kids to use for school work. The battery on my son's laptop seemed to be dying and got to the point where he could only use the laptop while it was plugged in. Recently, however, it has stopped powering on at all. The orange/yellow light on the front bezel comes on for as long as you hold down the power button, but the laptop doesn't actually do any normal startup procedures--no fans, other lights or post beeps. I tried using my son's power adapter on my daughter's laptop and it worked. I then tried using my daughter's power adapter on my son's laptop and got the same response (orange/yellow on the front bezel stays on solid for as long as you hold the power button down, but then goes out as soon as you let go of the button), so I don't believe the issue it the power cord/adapter. I did notice that the DC jack on my son's laptop appeared to be damaged (it was missing the plastic piece that surrounds the metal prongs). I ordered a replacement DC power jack and performed the swap myself. However, this does not seem to have resolved the issue. Since I had the laptop open and the battery disconnected during the DC power port swap, I took the opportunity to try to reset the laptop by holding the power button down for 30 seconds while I had the laptop unplugged and the battery out. I then put the battery back in, put the bottom cover back on and tried to power it up again. I got the same result (orange/yellow on the front bezel stays on solid for as long as you hold the power button down, but then goes out as soon as you let go of the button). Any thoughts on what else I might try to get my son's laptop up and running again?   Thank you,Z
Latitude 5500 Power Issue