OK, you have lots of great information and forumlae - you need direction - an overview.
When the spring in the toy ball launcher is compressed, we [theoretically] store an amount 1/2.k.x2
of energy in that spring. k is the spring constant, and x is the amount by which the spring is compressed [in metres of course]
When the launcher is "released" some of that energy is converted to kinetic energy of the ball - let's suppose all of it; I will explain later.
The ball now how kinetic energy given by 1/2.m.v2
m = mass, v = vel or speed
If that ball is released vertically in the upward direction, then as the ball rises, all that kinetic energy is converted to Gravitational Potential energy; given by m.g.h where these letters stand for the usual things.
You have most/all measurements to calculate any of those quantities and get what you want.
I hope you see that 1/2.k.x2
is effectively transformed into m.g.h
The real problem is - how efficient is the transformation of the energy from the spring.
If you were to compress the spring many times, with the release mechanism released, the spring will get warm. That means that during the compression/expansion of the spring some energy is converted to heat - probably not enough to worry about in a single "shot"
Secondly - and perhaps more importantly.
The spring itself has mass.
When released, one end remains stationary [against the "gun] while the other end accelerates to the same speed as the ball. If the mass of the spring is similar to the mass of the balls [or even worse; more] then the energy stored in the spring will be "shared" between the projected ball and the spring. Just how much of the stored energy is actually transferred to the ball? [I don't have an answer to that final question - it depends on the relative mass of the ball and spring]