The Star Trek: Khan comic had the Vengeance already under construction when Khan was brought to the project. Which is how I view it, as I think it would be impossible to design and build a new ship in less than a year.
That's reasonable, but I still want to blame the pointless hole in the saucer on him, just because he khan.
Anyway, building the Vengeance in a few months? Yeah, that's a bit too extreme. But designing and building those little drones that have their own phasers and torpedoes, and then do a suicide run when they're all out, seems much more feasible in that time, and they were kicking the Enterprise's butt over the moon.