Being ind-quasi-affine is a property of morphisms of schemes which is preserved under any base change, see More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-ind-quasi-affine}. Hence Descent, Lemma \ref{descent-lemma-descending-types-morphisms} applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{X \to S\}$ flat surjective morphism of affines. Say $X = \Spec(A)$ and $S = \Spec(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is ind-quasi-affine, in other words, $V$ is ind-quasi-affine. \medskip\noindent Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with $U = X$ and $R = X \times_S X$ and $s$, $t$, $c$ as usual. By Groupoids, Lemma \ref{groupoids-lemma-cartesian-equivalent-descent-datum} the pair $(V, \varphi)$ corresponds to a cartesian morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes. Let $u' \in U'$ be any point. By Groupoids, Lemmas \ref{groupoids-lemma-constructing-invariant-opens}, \ref{groupoids-lemma-first-observation}, and \ref{groupoids-lemma-second-observation} we can choose $u' \in W \subset E \subset U'$ where $W$ is open and $R'$-invariant, and $E$ is set-theoretically $R'$-invariant and an intersection of a nonempty family of quasi-compact opens. \medskip\noindent Translating back to $(V, \varphi)$, for any $v \in V$ we can find $v \in W \subset E \subset V$ with the following properties: (a) $W$ is open and $\varphi(W \times_S X) = X \times_S W$ and (b) $E$ an intersection of quasi-compact opens and $\varphi(E \times_S X) = X \times_S E$ set-theoretically. Here we use the notation $E \times_S X$ to mean the inverse image of $E$ in $V \times_S X$ by the projection morphism and similarly for $X \times_S E$. By Lemma \ref{lemma-affine-base-change} this implies that $\varphi$ defines an isomorphism \begin{align*} \Gamma(E, \mathcal{O}_V|_E) \otimes_R A & = \Gamma(E \times_S X, \mathcal{O}_{V \times_S X}|_{E \times_S X}) \\ & \to \Gamma(X \times_S E, \mathcal{O}_{X \times_S V}|_{X \times_S E}) \\ & = A \otimes_R \Gamma(E, \mathcal{O}_V|_E) \end{align*} of $A \otimes_R A$-algebras which we will call $\psi$. The cocycle condition for $\varphi$ translates into the cocycle condition for $\psi$ as in Descent, Definition \ref{descent-definition-descent-datum-modules} (details omitted). By Descent, Proposition \ref{descent-proposition-descent-module} we find an $R$-algebra $R'$ and an isomorphism $\chi : R' \otimes_R A \to \Gamma(E, \mathcal{O}_V|_E)$ of $A$-algebras, compatible with $\psi$ and the canonical descent datum on $R' \otimes_R A$. \medskip\noindent By Lemma \ref{lemma-sits-in-functions} we obtain a canonical ``embedding'' $$ j : (E, \mathcal{O}_V|_E) \longrightarrow \Spec(\Gamma(E, \mathcal{O}_V|_E)) = \Spec(R' \otimes_R A) $$ of locally ringed spaces. The construction of this map is canonical and we get a commutative diagram $$ \xymatrix{ & E \times_S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & & X \times_S E \ar[rd] \ar[ld]_{j''} \\ E \ar[rd]^j  & & \Spec(R' \otimes_R A \otimes_R A) \ar[ld] \ar[rd] & & E \ar[ld]_j \\ & \Spec(R' \otimes_R A) \ar[rd] && \Spec(R' \otimes_R A) \ar[ld] \\ & & \Spec(R') } $$ where $j'$ and $j''$ come from the same construction applied to $E \times_S X \subset V \times_S X$ and $X \times_S E \subset X \times_S V$ via $\chi$ and the identifications used to construct $\psi$. It follows that $j(W)$ is an open subscheme of $\Spec(R' \otimes_R A)$ whose inverse image under the two projections $\Spec(R' \otimes_R A \otimes_R A) \to \Spec(R' \otimes_R A)$ are equal. By Descent, Lemma \ref{descent-lemma-open-fpqc-covering} we find an open $W_0 \subset \Spec(R')$ whose base change to $\Spec(A)$ is $j(W)$. Contemplating the diagram above we see that the descent datum $(W, \varphi|_{W \times_S X})$ is effective. By Descent, Lemma \ref{descent-lemma-effective-for-fpqc-is-local-upstairs} we see that our descent datum is effective.