First, the reason the child is moving at all is that the potential energy at the top is converted into kinetic energy at the bottom. If the initial height is $h$ and the mass of the child is $m$, we have a final speed $v$ of
This is clearly independent of the path the child takes to get to the bottom, and so given that $h$ is the same in all cases, $v$ will also be the same. The child could be moving at a non-zero initial velocity, but this doesn't affect the result; the initial energy (and thus the final energy) will still be the same.
Also, the direction of velocity (a vector), though not the magnitude, depends on the angle of the slide; speed (a scalar) does not. Be careful that you don't confuse the two. The velocities are different; the speeds are the same.
As Floris pointed out, friction could make a different if it existed, but we're assuming that the slides are ideal and frictionless. Friction depends on the normal force exerted on the child, which does depend on the angle of the slide; the work done by friction would reduce the energy. However, we don't have to worry about this here.