Various answers have already been offered but I think you were already on the right track when you wrote:
"the instantaneous acceleration would actually affect the kinetic energy but its contribution would be taken care of when we consider the velocity after an small time interval."
In this question, velocity and mass are associated with energy. Acceleration (and other things like force) are more closely associated with POWER which is the flow or change of energy per unit time. Acceleration will affect velocity, and therefore energy, but it's not part of the instantaneous energy calculation.
A simple analogy, maybe, is that a ball falling downwards has a particular height above the ground at any moment in time. The downward velocity is also a property of the body, but it isn't part of the height measurement. But wait around a couple seconds, and the height will change in proportion to velocity. And then there will be a new height. And so on.