= Solution
Use the longer scale $T_s=(k_1E_0)^{-1}=T_f/\epsilon$ and $\tau_s=t/T_s=\epsilon\tau_f$. The equations become
$$
s_{\tau_s}=-s+(s+K-\lambda)c,\qquad\epsilon c_{\tau_s}=s-(s+K)c.
$$
At leading order the fast complex variable has equilibrated to the quasi-steady manifold. Therefore
$$
\boxed{c\simeq\frac{s}{s+K},\qquad\frac{ds}{d\tau_s}\simeq-\frac{\lambda s}{s+K}.}
$$
Matching from the <initial layer> gives $s(0)=1$ for this outer problem; its implicit solution is $\boxed{s+K\log s=1-\lambda\tau_s}$. The dimensional <substrate> loss is the Michaelis-Menten rate with maximum rate $k_2E_0$ and <Michaelis constant> $(k_{-1}+k_2)/k_1$.
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