Assume that NP $=$ DTIME($2^{\sqrt{n}}$), prove that DTIME($2^{\sqrt{n}}$) = DTIME($2^{n}$)

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 $ NP = DTIME(2^{n})$ .

The first case $ NP \subset DTIME(2^{n})$ , is trivial.

For the second case, let $ L \in DTIME(2^{n})$ and $ M$ be a deterministic $ TM$ that can decide it, let $ L_{pad} = \{\langle x,1^{2^{\sqrt{|x|}}} \rangle : x \in L\}$ . I am not sure how, using $ L_{pad}$ , I can reach a conclusion that $ NP = 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.