As the question stated, we know that $\{e_i\}$ doesn't form a basis for $l^{\infty}$. So how can we find a basis for $l^{\infty}$, no matter it is Schauder or Hamel basis.
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Every space which has a Schauder basis is separable. We have that $\ell^\infty$ is not separable, hence it can't have a Schauder basis. You can see a proof of this here, for example.
Using the axiom of choice, we have that every vector space has a Hamel basis - in particular, $\ell^\infty$ has a Hamel basis, but for infinite dimensional vector spaces we usually can't exhibit one.
The $\{e_i\}$ aren't a Schauder basis for $\ell^\infty$, but they are a Schauder basis for the subspace $c_0$ of the sequences that converge to zero. For a proof of this you can see my answer here.
