= Solution
The <half-line square-root potential well> is confining for $a>0$, which is assumed here. The hard wall imposes $\psi(0)=0$ and the smooth <turning point> is $x_{\rm t}=(E/a)^2$. With $u=a\sqrt x/E$, the WKB <action> becomes
$$
\int_0^{x_{\rm t}}k(x)\,dx
=\frac{2\sqrt{2m}E^{5/2}}{\hbar a^2}
\int_0^1u(1-u)^{1/2}\,du
=\frac{8\sqrt{2m}E^{5/2}}{15\hbar a^2}.
$$
The <hard-wall and soft-turning-point WKB quantization> condition therefore gives
$$
\boxed{E_n^{\rm WKB}
=\left[\frac{15\pi\hbar a^2}{8\sqrt{2m}}
\left(n+\frac34\right)\right]^{2/5},\qquad n=0,1,\ldots.}
$$
The coefficient $a$ here denotes the potential strength, not the <integration> limit in the generic quantization formula.
The <WKB approximation> is reliable in the interior where $|k'|/k^2\ll1$, but fails near the soft <turning point> where $k\to0$; a connection formula supplies its phase. The hard wall also requires its <boundary condition>, and $V'=a/(2\sqrt x)$ makes the local slow-variation condition fail sufficiently close to zero. Higher-energy states have a larger <action> relative to $\hbar$, with smaller relative endpoint regions, so the leading WKB spectrum is generally more reliable at large $n$ than for the <ground state>.
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