Answer
$h'=\frac{5h}{7}$
Work Step by Step
We use conservation of energy to obtain:
$mgh = \frac{1}{2}mv^2+\frac{1}{2}\frac{2}{5}mr^2\omega^2 \\mgh = \frac{1}{2}mv^2+\frac{2}{10}mr^2\omega^2 \\mgh = \frac{1}{2}mv^2+\frac{2}{10}mr^2\frac{v^2}{r^2} \\ gh = \frac{1}{2}v^2+\frac{2}{10}v^2\\ gh = \frac{7}{10}v^2 \\ v = \sqrt{\frac{10gh}{7}}$
Plugging in this value of v and using conservation of energy again, we find:
$\frac{1}{2}mv^2=mgh' \\ \frac{10}{14}gh=gh'\\ h'=\frac{5h}{7}$
