Here is a character-theoretic argument which shows that for each $n > 1$, whenever an involution $t \in {\rm SL}(2,2^{n})= G$ has $t = [a,b]$, then $a,b \in B = N_{G}(S)$, where $S$ is the unique Sylow $2$-subgroup of $G$ containing $t$.

In general, when $G$ is a finite group and $x \in G$, then the number of ordered pairs $(a,b) \in G \times G$ with $x = [a,b]$ is expressible as $\sum_{ \chi \in {\rm Irr}(G)} \frac{|G|\chi(x)}{\chi(1)},$ where ${\rm Irr}(G)$ is the set of complex irreducible characters of $G$. This formula was probably known to W. Burnside (in fact, the fact that $x$ is a commutator if and only if the sum is positive appears in Burnside's book, and was later important in the proof of the Ore conjecture).

Letting $T,S, B, G$ be as above, we note that $B$ is a Frobenius group of order $2^{n}(2^{n}-1)$ , and has $2^{n}-1$ irreducible characters of degree $1$ (each with $t$ in their kernel), and one irreducible character of degree $2^{n}-1$ taking value $-1$ at $t$. Hence the number of order pairs $(a,b) \in B \times B$ with $t = [a,b] $ is
$(2^{n}(2^{n}-1) [ (2^{n}-1) - \frac{1}{2^{n}-1}] = 2^{3n}-2^{2n+1}$.

On the other hand, $G$ has one irreducible character of degree $1$, the trivial character, one irreducible character of degree $2^{n}$ (which vanishes at $t$), $2^{n-1}$ irreducible characters of degree $2^{n}-1$, all taking value $-1$ at $t$, and $2^{n-1}-1$ irreducible characters of degree $2^{n}+1$, all taking value $1$ at $t$.

Hence the number of ordered pairs $(a,b) \in G \times G$ with $t = [a,b]$ is

$2^{n}(2^{2n}-1) [ 1 - \frac{2^{n-1}}{(2^{n}-1)} + \frac{2^{n-1}-1}{2^{n}+1}],$ which also turns out to be $2^{3n} - 2^{2n+1}$.

Hence all ordered pairs $(a,b) \in G \times G$ with $t = [a,b]$ actually lie in $B \times B$, so it is not possible to express $t = [a,b]$ where $\langle a,b \rangle = G.$