Died while updating

This is a hardware failure - there is a short circuit inside the system.

First thing to try replacing is the DC power jack (the jack to which the DC plug connects).  That should solve the issue.

If it doesn't, the mainboard is bad and will need to be replaced.

Thanks for the quick reply.

But it would be a heck of a coincidence that the jack failed when the computer was updating, considering the fact that the other end of the power supply was not plugged into the wall at the time.  And that the power supply system, including the jack has worked for 7-8 years.  I've got to believe it has to do with the battery dying in the middle of updating.

Any system that's 7-8 years old has exceeded its expected lifespan by a factor of two.  However, to check:

Unplug the system, remove or disconnect the battery and hold the power button for 30 sec.  If the update caused a problem, this will reset the system.

Plug it in (leave the battery out) and see if it'll start up.  If it will, you're set.  If you still see the LED on the adapter go out when you plug in the DC cable, there's a short somewhere - the DC jack is the first thing to replace.  Anything more than that on a 7-8 year old system means you should carefully evaluate the cost of a mainboard replacement (a few hundred dollars) against the cost of a new system.

OK, I did what you said and it's still dead.  A quick question:  In normal oeration, the computer will work with the battery out and the power supply connected? 

Unless something else is wrong, yes - you need to find the short and fix that.  If the DC jack doesn't do it, the mainboard is the faulty part.