If $T$ is the period of $SHM$,then write the period of kinetic and potential energy.

(T/2) In $SHM$,the displacement is given by $x = A \sin(\omega t)$.
The kinetic energy is $K = \frac{1}{2} k A^2 \cos^2(\omega t) = \frac{1}{4} k A^2 (1 + \cos(2\omega t))$.
The potential energy is $U = \frac{1}{2} k A^2 \sin^2(\omega t) = \frac{1}{4} k A^2 (1 - \cos(2\omega t))$.
Both kinetic and potential energy expressions depend on $\cos(2\omega t)$,which has an angular frequency of $2\omega$.
The period of $SHM$ is $T = \frac{2\pi}{\omega}$.
The period of kinetic and potential energy is $T' = \frac{2\pi}{2\omega} = \frac{T}{2}$.