Battery / Charge / CMOS (?) problem with Inspiron N4010 (14R)

My neighbors bought a 14R laptop for their daughter last Christmas.  They called me with a problem about it not starting up after it had been sitting around not in use for a while.  I thought that perhaps the battery had lost it's charge so asked them to try it with the laptop plugged in.  That worked, I think, b/c last night they brought it to me saying they couldn't get it to start even when plugged into the wall, and that when it was last working the battery wasn't holding a charge.

When I received it the 3rd light from the left on the left of the laptop was solid white, which means that the battery is charged.  I tried booting it and nothing happened - no beeps, no BIOS, nothing.  I then plugged into the wall and there was no difference.  I pulled out the battery and tried to boot with power from the wall only, and I still couldn't it to power on.  I tried all the previous steps again and the results were the same.  I reseated RAM, and again no change.  On a desktop, I'd start removing connections, but I really don't want to mess with taking the laptop apart.

So I start searching for answers and at some point, after another battery pull out to see if it had test buttons (of course not), when I pushed it back in and for the heck of it tried to boot, it booted to BIOS with a CMOS checksum error.  The 1st of the buttons on the left was solid white and I think that means the power button is on. The 3rd light was still solid white.  I went into setup and set the day and time then booted up to windows.  (I cannot log in to check for error logs b/c I don't have the blasted password.)

So, what's going on?  I've restarted and didn't need to go back to BIOS to reset day/time, so the CMOS battery appears to be charged.  The system is at the logon window, but now the 3rd light on the left (battery charge) is off, even though it has been operating only with wall power and has never been off until now.

Has anyone seen this?  Giving it back to them and essentially saying "cleared while testing" won't help matters as I know they will have the problem again.

Thanks for any assistance you can provide.