Let $d=dist(P,P^{\prime})\geq N=[d]$ be the intrinsic diameter of $L$. By connectedness, we can find $P_{1},\ldots P_{N}$ with $dist(P,P_{i})=i$. Now the intrinsic balls $\hat{B}(P_{i},\frac{1}{2})$ are disjoint, but each takes up a nontrivial amount of volume $\geq C^{-1}$. Thus

$$C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,$$

so there is an a priori bound on $N$, hence on $d$. ∎