It's a 2010 imac and when I bought it used it had ML installed.
I have a backup that's a few weeks old but every time I try and restore it I get read errors. Essentially I get the same error as I did on the drive that failed.
I put in another drive I had and tried the internet recovery, but it asks for my log in for the App Store, which I have no purchase under my account.
Okay, it sounds like your machine has had
this firmware update applied, so Internet Recovery is at least present. Normally, this would get you back to Lion or Mt. Lion that came with the machine. But since your machine came with Snow Leopard, it will not install Mt. Lion unless you have an AppleID tied to an App Store purchase of Mt. Lion.
Do you have access to any Mac anywhere that you could use the App Store app to login with your AppleID and buy ML for $20? Then go back to Internet Recovery on your machine and enter your AppleID when prompted during recovery.
At this point your options are buy a Snow Leopard disc from 1800MYAPPLE or figure out a way to access another boot drive for your machine or someone elses to get Mt. Lion purchased on your AppleID.