Discrete GPU not showing in System Report, what to look for

Just recently I discovered that I wasn't getting the GPU features I expected from my Macbook. After some investigation I discovered that the System Report is only showing the integrated Intel 4000 HD graphics, not the nVidia GT650M discrete GPU that came with the original system. I've also tried gfxCard status which shows only the Intel graphics. Oddly, when I try to switch to "integrated only" I'm not able to because Google Chrome Helper requires the discrete GPU (which, as you may recall, doesn't exist as far as OS X is concerned).

Additionally, for the last several months when rebooting it can take up to several hours before I'm able to actually log in. During that period the screen is completely black, although there are startup chimes. There are no errors I can find in the system logs and EtreCheck is not reporting anything other than com.adobe.ARMDCHelper.* (which seems unrelated).

It's as if the nVidia GPU was never there and never meant to be there. I have no idea how long it hasn't been working since there are no errors to indicate it isn't present and I'm not in the habit of periodically checking my system report just in case some of my hardware has disappeared. Could the "missing" GPU be related to the restart problems?

Is this something that might be fixed by resoldering? Or is it more likely the chip itself has failed due to thermal stress? Is there any way to tell? Any other possible cause?

Apple wants $566 to replace the logic board (my machine is "not one of the affected machines" a determination made by, I'm guessing, dowsing since they were not able to actual run any diagnostics) which really isn't all that bad, but when you add in the dodgy battery ( "Service Battery", 914 cycles -- which honestly seems a bit low for a machine that's been used daily for 5 years) and needing a larger SSD, we're starting to approach new machine territory. Spending $2-300 for a fix though seems reasonable -- more for the ability to reliably restart (so I can confidently apply system updates) than for unlocking greater GPU power -- if I have a reasonable expectation that it will help.

Thanks for any tips.