Proof of Space Hierarchy Theorem incompatible with Linear Speed Up Theorem for time
The Wikipedia proof is bogus.

As you have proven, the description for $L$ is incomplete. It should read:
$$
 L =  \{ (\langle M \rangle, 10^k) : M \mbox{ does not accept } x = (\langle M \rangle, 10^k) \mbox{ using space } \le f(|x|)
 \; \textbf{and time} \; \le 2^{f(|x|)} \}
$$

This corrects the proof because now $M$ (or $M'$) can still accept $x = (\langle M \rangle, 10^k)$ and take space $\le f(|x|)$. Actually, $(\langle M \rangle, 10^k) \in L$ has a quite reasonable explanation in that $M$ attempts to simulate $M$, which (as a subroutine) attempts to simulate $M$ again, and so forth and so on, until eventually $2^{f(|x|)}$ steps elapse and the whole procedure is interrupted. Hence, $M$ does not accept $x$ using space $\le f(|x|)$ and time $\le 2^{f(|x|)}$, although it does accept it taking space $\le f(|x|)$ (well, perhaps actually $O(f(|x|))$) and time $&gt; 2^{f(|x|)}$.

As a matter of fact, the purported algorithm for $L$ in the article explicitly states it runs its simulation with a $2^{f(|x|)}$ stopwatch. This is precisely the detail that is missing from the definition of $L$.