Dell Latitude E5440 does not power on

So your Dell is plugged in, but the battery does not seem to be charging. The power button is illuminated, but nothing happens when you press it. Perhaps the battery has been discharged so much that the motherboard power circuit has triggered to protect itself.

The following original source from here.

Which is summarized as follows:

1. Remove the battery and disconnect the power brick.
   
2. Hold the power button on for 10 seconds to fully discharge the power.
3. Next, while the battery is still disconnected, press and hold the power button while plugging in the power adapter. Hold the power button until the Dell splash appears.
4. Next fix the BIOS settings the defaulted when the battery hit 0%.

Finally insert the battery and monitor that it is taking a charge. You may need a new battery.