The ellipse has $a^2=144,b^2=169$, so its major axis is along the $y$-axis. Eccentricity of ellipse: $e^{\prime }=\sqrt{1-\frac{144}{169}}=\frac{5}{13}$. Foci of ellipse: $(0,\pm b{e}^{\prime })=(0,\pm 5)$.

The hyperbola $\frac{x^2}{16}-\frac{y^2}{{\lambda }^2}=-1$ rewrites as $\frac{y^2}{{\lambda }^2}-\frac{x^2}{16}=1$ — vertical transverse axis. Its foci: $(0,\pm \lambda \sqrt{1+16/{\lambda }^2})$.

Setting $\lambda \sqrt{1+16/{\lambda }^2}=5\Rightarrow {\lambda }^2+16=25\Rightarrow \lambda =3$.

Eccentricity of $H$: $e=\sqrt{1+\frac{16}{9}}=\frac{5}{3}$.

Length of latus rectum of $H$ $=\frac{2\cdot 16}{3}=\frac{32}{3}$.

$24(e+L)=24\left(\frac{5}{3}+\frac{32}{3}\right)=24\cdot \frac{37}{3}=296$.