= Solution
Let $F_n(x)=n^{-1}\sum_i\mathbf1_{\{X_i\leq x\}}$. The <Kolmogorov-Smirnov statistic> is $D_n=\sup_x|F_n(x)-F(x)|$, often reported as $\sqrt nD_n$. Under the simple null, the <probability integral transform> makes $U_i=F(X_i)$ independent uniform variables on $[0,1]$. Strict monotonicity and continuity identify the empirical distribution under this change of coordinates, so
$$
D_n=\sup_{0\leq u\leq1}|G_n(u)-u|.
$$
Its null law is therefore independent of $F$.
For the composite null, the joint density is $\theta^{-n}\prod_i\mathbf1_{\{0\leq x_i\leq\theta\}}$. Its parameter dependence is only through $M=\max_iX_i$, proving sufficiency by the <Fisher-Neyman factorization theorem>; the larger statistic $(J,M)$ is sufficient too. Ties have probability zero. For fixed $j$, the density of $(J=j,M=\xi)$ is $\xi^{n-1}/\theta^n$, $0<\xi<\theta$, because the other coordinates range independently over $[0,\xi]$. Dividing the joint density by this marginal gives conditional density
$$
\xi^{-(n-1)}\prod_{i\ne j}\mathbf1_{\{0\leq x_i\leq\xi\}}.
$$
Thus the remaining observations are conditionally independent $U[0,\xi]$.
For $n\geq2$, discard the maximum and apply the <Kolmogorov-Smirnov test> to the $n-1$ ratios $X_i/M$, $i\ne J$, against $U[0,1]$. \b[Use the ordinary null critical values for sample size $n-1$.] Its conditional, hence unconditional, level is independent of $\theta$. Keeping the fitted maximum and using the ordinary $n$-sample critical values would not have that justification.
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