I don't understand how the volume of 0.200 M NaOH needed to neutralize 250.0 mL of 0.010 M HCl (0.0125 mL) is the same volume of 0.200 M NH3 that would be needed to neutralize 250.0 mL of 0.010 M HCl.
Well it's because the strength of an acid or base is a measure of it's dissociation in water. But for the the reaction of an acid with a base the equilibrium lies completely to the right as the products (salt and water) are much more stable than the acid and base.
You're trying to compare the dissociation of a species in water with the equilibrium constant of an actual reaction. I understand why, but think of them as two different situations.