So! The good news is that the ice on top of the tank was only skim ice. That means it hadn't been frozen that long (basically, we could trace that back to when the toilet backed up). But, this is what was the problem (see pictures). I don't know if you can tell from this picture or not, but the input line (that runs from the house to the tank) was FULL and FROZEN SOLID.
Ah-ha! Problem found!