I think that the way the directions are written is a bit confusing. It has a K2tog and then *k8, K2tog; repeat from * Mathematically, that just doesn't work. To me, the way it is written is for a multiple of 10+2. I agree with Ingrid: K2tog, K8 (or K8, K2tog, it doesn't matter) around, then K2tog, K7, etc.
And regarding one edge having a K2tog and one edge having a K8 (one slanted decrease edge and one straight edge), that's how it should be because then the decreases are evenly spaced around - otherwise you'd have 2 decreases side by side, which isn't the case over the rest of the hat.