Even if missiles were out of the equation, where was that sword they used all along up to that point?!
If they had martial arts masters running the robots (as evidenced by the "tryouts" scene), they should have had swords and other weapons to simply make sushi out of the Kaiju!
I really enjoyed the movie, but that doesn't mean the characters were all that smart. Negative points on the movie's score for the characters' intelligence level serving the plot rather than the other way around.
"I'm losing this battle. Oh yeah - I've got a sword on this arm. AND on this arm! Guess I will use them in all subsequent battles!" - figured out 2/3 the way through the film...