You'll need to replace the mainboard - when the LED on the adapter dies on connection to the system, the mainboard is faulty.
But it runs just fine off the battery.
Could it be something as simple as replacing the connector?
It could be - depends on the system and would require a professional evaluation to tell for sure.
Replacing the jack will run $150-200 to have it done correctly.
I was going to do it myself.
I've got the laptop stripped down and the solder joints on the connector look pretty ugly. Like something that would have never passed QC at the factory I used to work for.