Hi Brian,
First, based on similar experiences with an 2010 iMac 27, the overall problem may not be recoverable at a reasonable price.
You can try Target Disk Mode to access the iMacs hard drive if you have another Mac. If the drive is working, it will show up on the desktop of another Mac as an external hard drive.
Transfer files between two Mac computers using target disk mode - Apple Support
That did not work on my 2010 iMac. It seemed the drive controller or other important bit of kit was somehow affected. Fortunately I had two independent backups to external drive so I lost nothing.
If:
- Target Disk does not work,
- you have no external backups,
- but the drive works
the last resort is to open the computer, extract the hard drive and attach it to a universal USB adapter or a drive docking station and access the data from another computer that way.
"Not dead" video cards for 2009-2011 iMac are hard to find, hard to install, and can cost far more than any sane person would pay for a fully-working, minty, working 2011 iMac. I do not see that as a viable option. You can replace the card with an expensive equivalent and still not have a working computer.