You're most of the way there, a couple minor fixes, which you actually almost caught yourself: The force affects ONLY the vertical velocity. Impulse is, as you noted, split into two components. The vertical (incidentally, that should be 0.002s + 0.001s in that calculation, those triangles on the graph are each 2ms, and the centre is 1ms), and the horizontal impulse is 0. I'm going to use subscripts for the directions, i.e., vj is vertical velocity.
So, what you calculated is Ij
As you noted, the horizontal impulse Ii is zero.
The vertical velocity change equation is Ij = m(Vj - Vj0)
The horizontal velocity change is Ii = 0 = m(Vi - Vi0)
Vj = 2.50m/s * cos60
Vi = 2.50m/s * sin60
The importance of that is that when you're using the vertical impulse, it will only affect the vertical velocity Vj, Vi is only going to be affected by horizontal impulse (here zero).
The second thing to watch for is to remember which way your negative is. If Vj starts as being positive, then the j direction is DOWN, not up, and a negative Vj implies a ball that is moving upwards.
EDIT: Oh, I may have swapped the directions on you, I was using j as the vertical, but you can swap i and j around as long as you're consistent.
Ibid: Most prolific author in academia