The spring stiffness of each part is twice the the original one, that can be considered, the two parts connected in a series. The stiffness of two springs connected in a series is k1xk2/(k1+k2) that brings us to k/2 in the case k1=k2=k. So, if we consider the two parts as two springs with a stiffness of k, after joining them to the original one, its stiffness is half of k.

This calculation should enable you to work out the stiffness of the spring after dividing it into two equal parts, as far as your question goes.

This calculation should enable you to work out the stiffness of the spring after dividing it into two equal parts, as far as your question goes.