Let $G$ be a group with finite presentation $\langle S\mid R \rangle$. For every reduced word $w$ in the free group $F(S)$ denote $|w|$ its length. For every $w \in F(S)$ that represents the identity in $G$ define $A(w) = A$ as the minimum number such that we can write $$ w = w_1r_1^{a_1}w_1^{-1}\cdots w_n r_n^{a_n} w_n^{-1} $$ with $n \in \mathbb{N}, w_1,\ldots,w_n \in F(S), r_1,\ldots, r_n \in R, a_1, \ldots, a_n \in \mathbb{Z}$ and with $A$ that verifies $$ A = |a_1| |r_1|^2 + \cdots + |a_n| |r_n|^2\\ |w_1| + \ldots + |w_n| \leq A $$
Gromov in "Hyperbolic Groups" p.103 Theorem 2.3.A proved the following isoperimetrically inequality for hyperbolic groups.
Theorem Let $G$ is an hyperbolic group finitely generated by $S$. Suppose the Cayley graph $Cay(G,S)$ is $\delta$-hyperbolic. Consider the set $W_d \subseteq F(S)$ of reduced words of length at most $d$ and that represent the identity in $G$. If $d \geq 8(\delta+2)$, then $\langle S \mid W_d \rangle$ is a presentation of $G$. Moreover if $w \in F(S)$ is the identity in $G$, then $$ A(w) \leq 27d^2|w| $$
As a corollary he stated that the word problem for hyperbolic groups is solvable.
My question is: given $\langle S\mid R\rangle$ a finite presentation of an hyperbolic group, what is the algorithm that solves the word problem? It is obvious that once I know the set $W_d$ then I can just check every expression as in the definition of $A$ and give an answer, but how can I determine $W_d$?