For simplicity take this problem in two parts. (You can do this in one step, but it's messy.)
First part: The ball falls to the level of the top of the spring (ie. falls 62 cm).
How fast is it moving just before it hits the spring? I've got a coordinate system at the point where the ball is dropped and I'm calling +y upward.
(Negative since it's falling downward.)
Second part: The ball compresses the spring.
This is a work - energy problem. I'm defining the 0 level for the gravitational potential energy to be at the top of the uncompressed spring, and the 0 level for the spring potential energy to be at the top of the uncompressed spring, and in both cases I'm taking +y upward. There are no non-conservative forces at work here, so:
The ball starts at the top of the spring ( ) with the speed from the last part of the problem, and ends at with a speed of 0. So: