Apart from the above answer, I suggest that "I ran" has two different meanings. First, a literal meaning, one must run to fly a kite. Second, a figurative meaning, running sometimes allows a person to feel free or escape. That said, bullgatortail's answer proves to be very analytical (meaning amazing)!
Amir has just made a great breakthrough with his nephew, Sohrab: Sohrab has smiled while kite-flying, and Amir asks if he can run the kite for him just as Hassan had done for Amir when they were youths. For Amir, it is the ultimate pleasure: running the kite for Sohrab means a new beginning for them, and it is a way of making amends for his past sins against Hassan. Now it is Amir who relegates himself to a more subservient role, yet another step toward atonement for Amir.