First we observe that the map $\Pic(X) \to \Pic(X_{k'})$ is injective by Varieties, Lemma \ref{varieties-lemma-change-fields-pic}. Hence we have to show the map in the lemma is surjective. Let $\mathcal{L}$ be an invertible $\mathcal{O}_{X_{k'}}$-module which has order dividing $n$ in $\Pic(X_{k'})$. Choose an isomorphism $\alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_{X_{k'}}$ of invertible modules. We will prove that we can descend the pair $(\mathcal{L}, \alpha)$ to $X$. \medskip\noindent Set $A = k' \otimes_k k'$. Since $k'/k$ is purely inseparable, the kernel of the multiplication map $A \to k'$ is a locally nilpotent ideal $I$ of $A$. Observe that $$ X_A = X \times_{\Spec(k)} \Spec(A) = X_{k'} \times_X X_{k'} $$ comes with two projections $\text{pr}_i : X_A \to X_{k'}$, $i = 0, 1$ which agree over $A/I$. Hence the invertible modules $\mathcal{L}_i = \text{pr}_i^*\mathcal{L}$ agree over the closed subscheme $X_{A/I} = X_{k'}$. Since $X_{A/I} \to X_A$ is a thickening and since $\mathcal{L}_i$ are $n$-torsion, we see that there exists an isomorphism $\varphi : \mathcal{L}_0 \to \mathcal{L}_1$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-torsion-pic-thickening}. We may pick $\varphi$ to reduce to the identity modulo $I$. Namely, $H^0(X, \mathcal{O}_X) = k$ implies $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology} and $A \to k'$ is surjective hence we can adjust $\varphi$ by multiplying by a suitable element of $A$. Consider the map $$ \lambda : \mathcal{O}_{X_A} \xrightarrow{\text{pr}_0^*\alpha^{-1}} \mathcal{L}_0^{\otimes n} \xrightarrow{\varphi^{\otimes n}} \mathcal{L}_1^{\otimes n} \xrightarrow{\text{pr}_0^*\alpha} \mathcal{O}_{X_A} $$ We can view $\lambda$ as an element of $A$ because $H^0(X_A, \mathcal{O}_{X_A}) = A$ (same reference as above). Since $\varphi$ reduces to the identity modulo $I$ we see that $\lambda = 1 \bmod I$. Then there is a unique $n$th root of $\lambda$ in $1 + I$ (Algebra, Lemma \ref{algebra-lemma-lift-nth-roots}) and after multiplying $\varphi$ by its inverse we get $\lambda = 1$. We claim that $(\mathcal{L}, \varphi)$ is a descent datum for the fpqc covering $\{X_{k'} \to X\}$ (Descent, Definition \ref{descent-definition-descent-datum-quasi-coherent}). If true, then $\mathcal{L}$ is the pullback of an invertible $\mathcal{O}_X$-module $\mathcal{N}$ by Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Injectivity of the map on Picard groups shows that $\mathcal{N}$ is a torsion element of $\Pic(X)$ of the same order as $\mathcal{L}$. \medskip\noindent Proof of the claim. To see this we have to verify that $$ \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi = \text{pr}_{02}^*\varphi \quad\text{on}\quad X_{k'} \times_X X_{k'} \times_X X_{k'} = X_{k' \otimes_k k' \otimes_k k'} $$ As before the diagonal morphism $\Delta : X_{k'} \to X_{k' \otimes_k k' \otimes_k k'}$ is a thickening. The left and right hand sides of the equality signs are maps $a, b : p_0^*\mathcal{L} \to p_2^*\mathcal{L}$ compatible with $p_0^*\alpha$ and $p_2^*\alpha$ where $p_i : X_{k' \otimes_k k' \otimes_k k'} \to X_{k'}$ are the projection morphisms. Finally, $a, b$ pull back to the same map under $\Delta$. Affine locally (in local trivializations) this means that $a, b$ are given by multiplication by invertible functions which reduce to the same function modulo a locally nilpotent ideal and which have the same $n$th powers. Then it follows from Algebra, Lemma \ref{algebra-lemma-lift-nth-roots} that these functions are the same.