= Solution
Write $N=|A|$. For $N>1$, the unmarked <computational basis> vectors in $A$ give the normalized vector
$$
\boxed{|\omega\rangle=\frac1{\sqrt{N-1}}\sum_{x\in A\setminus\{a\}}|x\rangle
=\frac{\sqrt N|\psi_A\rangle-|a\rangle}{\sqrt{N-1}}.}
$$
Its norm is one and $\langle a|\omega\rangle=0$, so these two vectors form an <orthonormal basis> for $\mathcal V$. Splitting the <uniform superposition state> into its marked and unmarked terms gives
$$
|\psi_A\rangle=\frac1{\sqrt N}|a\rangle+\sqrt{\frac{N-1}{N}}|\omega\rangle,
\qquad\boxed{\theta=\arcsin\frac1{\sqrt N}.}
$$
The nonnegative coefficients select the specified angle in $[0,\pi/2]$.
There is one degenerate boundary case: if $N=1$, then $|\psi_A\rangle=|a\rangle$ and $\mathcal V$ is one-dimensional. No vector in $\mathcal V$ can complete $|a\rangle$ to a two-element <orthonormal basis>. Thus the printed two-dimensional description needs $N>1$. For $N=1$ the search itself is already solved, with $\theta=\pi/2$ and no query.
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