Throwing more koji at a batch of sake isn't going to increase the alcohol content, enzymes just don't work that way. When made properly, koji is so enzyme rich that adding just 25% of your total grist as koji supplies more than enough diastatic power to break down all of the rice in your moromi. In fact, part of the reason for pasteurizing sake has to do with deactivating the remaining enzymes after all of the available amylose has been hydrolyzed.
What you're really asking, part one: So, why doesn't the rice break down completely?
Answer: For the same reason malt doesn't. Rice isn't made of just starch, it contains other proteins and minerals that make up the cell walls and give the grain its structure. Just like spent malt, sake lees are made up of the bits of the rice kernels that no amount of enzymes supplied koji is going to be able to break down. Those sake lees have given up all their goods, just like the spent grain has in a batch of beer.
What you're really asking, part two: How do I achieve a higher alcohol content in my sake?
Answer: Genshu (undiluted) sake has an alcohol content of up around 20% ABV. That's as high as the yeast can tolerate before they die by being poisoned by their own waste (alcohol). If you want it to be any higher (whyyyy?), adding more koji or yeast isn't going to get you there. This is where fractional freezing or distillation comes in.
I'd go for the freezing, if I were going to do it. Sakejack, anybody?