This is most likely a mother board problem. The best way to tell is to go into BIOS and see if it recognizes the power adapter properly, you can also use the built in Dell test to verify both the battery and the ac adaptor. If the BIOS shows that an unknown or a 1 watt power adapter is plugged in then this is definitely a mother board problem. The other symptom you have, where the blue light on the power cable turns off, also indicates a mother board problem.
Start with the laptop turned off and turn it on, as soon as you see the Dell logo hit F12 for diagnostics or F2 for Bios. The BIOS is divided into several sections, look at the top for information about the processor and power to see how the power adapter is being detected. Diagnostics will run all tests or you can select the component to test, there is a results tab where you can see if the test passed or failed.