Welcome to the Community,
Try to release flea power.Remove any peripherals connected to the system.Disconnect the AC adapter,remove battery,press and hold the power button for 30 seconds.Reconnect only the battery and AC adapter,turn on system,after it loads to windows,shut down the system and check if the power light goes off.
If the issue persist remove the battery and turn on and shut down system only with AC adapter connected (without the battery). Confirm if the power light goes off,when you shut down the system,while using it only with the AC adapter.
I have the same problem but none of these tips seemed to work. I have also updated all drivers and BIOS on the system. It is a fresh install of Windows 7 Pro on a Dell Inspiron N5050.
Well, I have solved my own problem.
For those who have this issue and the above suggestions didn't work...
I took the laptop apart and changed out the CMOS battery. After putting it back together, everything worked perfectly. The machine now shuts down completely.
I have tried the two suggestions but the light is still on when I put the battery back in on both attempts. It used to go out before. Is there anything else I should try?
PS: I forgot to confirm that the power light does go off when only the AC adaptor is connected. It only stays on when the battery is inserted.
I should add to my last reply that after your suggested solution and trying without a battery the power light DOES go off with only the AC adapter being used NO battery !