= Solution
Let $E_\ell$ be the event that the block beginning at position $\ell$, $1\leq\ell\leq k+1$, is a spalindrome. Each has probability $(10)_k/10^{2k}$. If two starting positions differ by $d<k$, the first half of the later block contains the middle digit of the earlier palindrome twice, contradicting distinctness. Thus the only possible overlap is $E_1\cap E_{k+1}$.
That intersection consists of words $A\,A^{\rm rev}A$, where $A$ has $k$ distinct digits, and hence has probability $(10)_k/10^{3k}$. This <overlap structure of digit spalindromes> makes inclusion--exclusion give
$$
\boxed{\mathbb P\left(\bigcup_{\ell=1}^{k+1}E_\ell\right)
=(k+1)\frac{(10)_k}{10^{2k}}
-\frac{(10)_k}{10^{3k}}.}
$$
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