KotR is way too potent to use on something as weak as Godo. Godo has, like, 10,000 HP, and KotR does almost 120,000 damage. I'm pretty sure something like Comet2 can wipe out Godo in one hit by that time. In fact, I'm prettty sure by the time you get KotR, you will be able to kill Godo easily with just physicals. So KotR itself, let alone mime, is a completely unnecessary measure.

The good strategy would be to use something like Osmose against him, to drain him of MP, so he won't use Cure2. Big Guard is also a good idea. Well Big Guard is a good idea in any boss fight. Also remember to give her your best equipment and materia since she's the only one fighting. Also, maybe it's just that he had ran out of MP just at the right moment, but he didn't seem to use Cure2 when I used a damage-inflicting item...maybe some sort of bug, I don't know. But if you equip yourself properly, or just cast Big Guard (which can be obtained from a monster on the shores near Costa Del Sol by the way, though you gotta manipulate the dude, I think, not sure), only the Demi attack should be a problem, and if you suck up his MP, you should do well.