What can be done to fix a 2011 iMac that will not boot past the white screen?
Here is what you know from your excellent details:


The keyboard isn't working since all of the shortcuts get handled from the firmware in the Mac.
The Mac can't find any bootable OS X due to corruption or hardware failure


Why the keyboard isn't working isn't yet clear. I would disconnect everything including the power and then remove both RAM sticks - noting which was on the top and which was on the bottom.

Next, plug the Mac in and power it up with nothing connected. You should get three beeps indicating the POST (power on self test) detected no RAM. If you have severe hardware problems, it won't know the RAM is missing and you'll know you need to get inside to fix it.

Assuming this works, power it off after waiting to see if it boots in 5 minutes time (really wait that long for the Apple logo to show up). While you are waiting listen carefully for rhythmic noised from the hard drive. It shouldn't keep seeking past the first 45 seconds and if so - you will know the controller or the drive is faulty. It should spin quietly after a few minutes or be reading the drive if it's corrupt and the hardware / OS is trying to fix things.

Next - power it off by the button and put the "top" piece of RAM in the "bottom" slot and repeat. Lastly - if you get to the white screen at this point, try another keyboard (command-s or shift held down are the best options here for troubleshooting) to see if you can isolate the failure further.

I tried to make some assumptions about what is wrong and you can edit the question or ask another with the step that fails referring to this question if you want to isolate things further before attempting a repair.