I tried using the padding argument to prove such a thing (as it appeared in Arora's book), but I am not sure how this technique will help me here. I am trying to get to a contradiction to the Time Hierarchy Theorem.
After the assumption, I want to prove that $\mathsf{NP} = \mathsf{DTIME}(2^{n})$.
The first case $\mathsf{NP} \subset \mathsf{DTIME}(2^{n})$, is trivial.
For the second case, let $L \in \mathsf{DTIME}(2^{n})$ and $M$ be a deterministic TM that can decide it, let $$L_{\mathrm{pad}} = \left\{\left\langle x,1^{2^{\sqrt{|x|}}} \right\rangle : x \in L\right\}.$$ I am not sure how, using $L_{\mathrm{pad}}$, I can reach a conclusion that $\mathsf{NP} = \mathsf{DTIME}(2^{n})$, I feel like I am missing something and that my method is not in the right direction, or it is lacking an extra detail.