May I refer you to theorem $8.32$ on page $146$ in Metric Spaces of Non-Positive Curvature
In the last paragraph, why is it the case that $H$ has finite index implies there is a constant $\mu$ such that every vertex of the Cayley graph lies in a $\mu$-neighborhood of $H$?
Let $G$ be the group that contains $H$ with finite index. By definition of finite index, there exists a finite subset $\{g_1,…,g_K\}$ of $G$ such that $$G = H g_1 \cup H g_2 \cup \cdots \cup H g_K $$ Given any $g \in G$, there exists $h \in H$ and $k \in \{1,…,K\}$ such that $g = h g_k$, so $d(g,h) = d(h^{-1} g,\text{Id}) = d(g_k,\text{Id})$, and therefore $$d(g,H) \le \mu = \text{Max}_{k=1}^K d(g_k,\text{Id}) $$