When do contributions of $\pi_*(L_{K(n)}S^0)$ to $\pi_*(S^0)$ stabilize? As a word of warning: even once $\pi_m S \to \pi_m L_{E(n)} S$ becomes injective, it can fail to be surjective indefinitely without violating the convergence theorem. So, even if you had some lower bound on n and a computation of $\pi_m L_{E(n)} S$, it will be hard to sort out what information is "real" and what is chromatic crosstalk.