Impacts are complex physics problems. Amusing the ball has mass, it cannot change velocity instantly, so its speed just before and just after the impact must be the same.

At the exact moment of impact, the ball will be compressed, so the bottom side of the ball will slow down before the top of the ball slows down. There won't be a single speed that describes all parts of the ball. The change in velocity will travel from the bottom of the ball to the top of the ball at the speed of sound in the ball material. The axis will experience a similar deformation. Depending on the material properties, the material may also permanently deform at the contact point, taking energy away from the velocity.

Ignoring the contact situation and assuming the axis and ball will move at the same velocity together, without the ball bouncing, the velocity after impact will depend on the mass and inertia of the axis, which can be calculated from it's length, and the ratio between the mass of the axis and the ball, which is unknown since the mass of the ball was not given.
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