In this case, this follows because $E(n)$-localization is a smashing localization. Specifically, $$ L_{E(n)}(X) = L_{E(n)}(\mathbb{S}) \wedge X $$ for any $X$. In particular, this means that the smash product of any spectrum with an $E(n)$-local spectrum is already $E(n)$-local.
The fact that $E(n)$-localization is smashing was one of the Ravenel conjectures; specifically, conjecture 10.6 in his paper "Localization with respect to certain periodic homology theories". I don't have a copy of his orange book handy and so can't give you a reference for the theorem statement.
As a result, as the Morava $E$-theories are $E(n)$-local already, so are the smash products $E \wedge X$ in your equation.