= Solution
Take pumping length $p=1$. If a nonempty word $w\in\{0,1\}^*$, pump its first symbol; every pumped word remains in $\{0,1\}^*$. Otherwise write $w=u2v$ with $v\in L$. If $u\ne\varepsilon$, pump the first symbol of $u$, leaving a word of the same form. If $u=\varepsilon$, pump the initial $2$: deleting it leaves $v\in\{0,1\}^*$, while retaining one or more copies lets the last pumped $2$ serve as the separator. Thus every required pumped word belongs to $\widehat L$.
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