If you have a second power adapter try to turn the laptop on with it. The most common problem on Dell Mother boards is the power handling components. If you are able to get the laptop to power on, I suggest checking the BIOS to see what it shows for AC adapter and battery. There is also a diagnostic system that can test different components. To get to BIOS or diagnostics, hit F12 when you see the Dell splash screen and you will see a menu that will take you to Bios or Diagnostics.
If your computer is still in warranty then call Dell tech support and they will dispatch a tech to replace the part. Out of warranty, they can dispatch a tech but you will have to pay for the service.