Unless we make an additional assumption not stated in the problem definition, $v\sin{\theta}$ does not necessarily $= \sqrt{2gh}$. The equations you are using do not make this assumption, so they are arriving at a different result.

The $v\sin{\theta} = \sqrt{2gh}$ statement only holds if we assume that the basket is at the top of the ball's arc, which was not one of your explicit assumptions. The equations you are using do not seem to account for this assumption either, since $t_b$ refers to the time spent on the complete arc, not just the part where it reaches height $H$.

With our additional assumption, we could assume that all of the ball's initial kinetic energy was converted into potential energy by the time it rose by distance $h$. Then, the proof simply becomes:

\begin{equation}
\begin{split}
\Delta K &= \Delta P \\
\frac{1}{2}m(v\sin{\theta})^2 &= mgh \\
v\sin{\theta} &= \sqrt{2gh}
\end{split}
\end{equation}

Where $\Delta K$ is the kinetic energy lost over distance $h$ and $\Delta P$ is the potential energy gained over the distance $h$. 

Without our additional assumption, the $K$ converted to $P$ over distance $h$ could account for any fraction of the initial $K$. 

So, let

\begin{equation}
\begin{split}
K &= K_h + K_a \\
P &= P_h + P_a \\
K_h &= P_h \\
K_a &= P_a
\end{split}
\end{equation}

where:

* $K$ is the total kinetic energy the ball loses while going up.
* $P$ is the total potential energy the ball gains while going up.
* Subscript $_h$ refers to quantities corresponding to motion from height $L$ to height $H$, over distance $h$.
* Subscript $_a$ refers to quantities corresponding to motion at heights above $H$.

It follows that:

\begin{equation}
\begin{split}
K &= P \\
K_a + K_h &= P_a + P_h \\
K - K_a &= K_h \\
K - K_a &=  P_h \\
\frac{1}{2}m(v\sin{\theta})^2 - K_a &= mgh \\
v\sin{\theta}-\sqrt{\frac{2}{m}K_a} &= \sqrt{2gh}
\end{split}
\end{equation}


If $K_a$ is non-zero (which will happen if the ball reaches a height _above_ the basket), then the statement $v\sin{\theta} = \sqrt{2gh}$ is un-provable.