The box slows steadily until it stops.
Let us understand why this should be the case.
We know that the Tension Force in the rope is greater than Friction and hence is responsible for accelerating the box.
As soon as the rope breaks, the Tension Force becomes 0 and the box is left with only the Friction acting on it.
Since the friction takes over immediately, the box begins to slow down at a steady pace until it stops completely.