So we can divide the time period in two parts. first $T_{1}$ and second $T_{2}$

$T_{1}$ is half of the time period of a full SHM i.e. $T_{1}=\pi \sqrt{\frac{m}{k}}$

$E=\frac{1}{2} m v_{\max }^{2}$

$\Rightarrow v_{\max }=\sqrt{\frac{2 E}{m}}$

and $T_{2}$ is the time during which the particle remains in air when it is thrown upwards with velocity $v_{\max }=\sqrt{\frac{2 E}{m}}$

$0=\sqrt{\frac{2 E}{m}} t-\frac{1}{2} g t^{2} \quad$ since $s=u t-\frac{1}{2} g t^{2}$

$\Rightarrow T_{2}=2 \sqrt{2 E / m g^{2}}$

So total time time period $T=\pi \sqrt{\frac{m}{k}}+2 \sqrt{\frac{2 E}{m g^{2}}}$