Pages 75-81 of Appendix A of ``On the theory and applications of differential torsion products'', Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-construction for simplicial augmented algebras over a commutative ring $R$. Not the answer to your question, but if I remember rightly, it should lift to an answer when suitably specialized; more precisely, the chain homotopy of Lemma A.16 (up to signs coming from variant choices) should specialize to one coming from a contracting homotopy as desired. When $G$ is a group regarded as a constant simplicial group, Lemma A.15 lifts to give an isomorphism between $WG$ and the simplicial set $E_*G$ whose realization is $EG$.

Pages 75-81 of Appendix A of On the theory and applications of differential torsion products, Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-construction for simplicial augmented algebras over a commutative ring $R$. Not the answer to your question, but if I remember rightly, it should lift to an answer when suitably specialized; more precisely, the chain homotopy of Lemma A.16 (up to signs coming from variant choices) should specialize to one coming from a contracting homotopy as desired. When $G$ is a group regarded as a constant simplicial group, Lemma A.15 lifts to give an isomorphism between $WG$ and the simplicial set $E_*G$ whose realization is $EG$.