Inspiron 3584 nearly six years old...

TRUE NOVICE, but I enjoy tinkering with these things

Recently, I purchased a 3550 to replace the 6-year-old 3584 soley for my personal activities but I could really use a 2nd laptop to have multiple screens during stock trading hours. 

I misdiagnosed the 3584's behavior and just replaced the power adapter and DC power jack cable to no benefit except to eliminate them as responsible.

I'm imagining I should have started with a new battery, but I didn't because I was thinking I'd replaced it a couple years ago. (need to keep a log)

Symptoms: LED next to power cable was lighting up when plugged in but no longer does at all. 

Secondly, the LED on the Caps Lock button will sometimes light up for several minutes but the screen only illuminates enough to glow, but the Dell logo is no longer appearing at all, no matter how long the light on Caps remains lit before eventually going out. 

Does this sound like the battery is no longer even acting as a conduit?

Thanks for whatever experience folks can draw from!

Robert