= Solution
Use the simple-root form of the <Hensel lemma>: for $F\in\mathcal O_K[T]$ and $\bar a\in k_K$ satisfying $\overline F(\bar a)=0$ and $\overline{F'}(\bar a)\ne0$, there is a unique $a\in\mathcal O_K$ with $\bar a$ as its reduction and $F(a)=0$.
Let $d=[\ell:k_K]$. The <residue field> $k_K$ is finite, so $\ell/k_K$ is separable, and the <primitive element theorem> gives $\ell=k_K(\bar\alpha)$. Let $\bar F$ be the monic irreducible polynomial of $\bar\alpha$, of degree $d$, and choose a monic coefficient lift $F\in\mathcal O_K[T]$. Irreducible reduction implies that $F$ is irreducible over $K$: any monic factorization over $K$ has integral coefficients, since all its roots are integral over the <valuation ring>, and would reduce to a factorization of $\bar F$.
Adjoin a root $\alpha$ and put $L=K(\alpha)$. Then $[L:K]=d$ and $\alpha\in\mathcal O_L$. Its reduction satisfies $\bar F$, so $k_L$ contains a copy of $\ell$ over $k_K$. For finite extensions of non-Archimedean <local fields>, the fundamental degree relation is
$$
[L:K]=e(L/K)[k_L:k_K].
$$
The residue degree is at least $d$, while the left side equals $d$, hence
$$
\boxed{e(L/K)=1,\qquad k_L\cong\ell,\qquad [L:K]=[\ell:k_K].}
$$
This constructs the required <unramified extension>.
For uniqueness, let $L'/K$ be another unramified extension with a chosen identification $k_{L'}\cong\ell$. The element corresponding to $\bar\alpha$ is a simple residue root of $F$, because finite fields are perfect. The <Hensel lemma> in $L'$ gives a root $\alpha'\in\mathcal O_{L'}$ with this reduction. Since $F$ is irreducible of degree $d$, $[K(\alpha'):K]=d=[L':K]$, so $L'=K(\alpha')$. Sending $\alpha$ to $\alpha'$ yields a $K$-isomorphism $L\to L'$, compatible with the prescribed residue identification. Thus
$$
\boxed{\text{the unramified lift of }\ell/k_K\text{ is unique up to }K\text{-isomorphism}.}
$$
This proves the <existence and uniqueness of unramified local extensions> using the stated version of Hensel's lemma, without assuming the classification in advance.
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