  For the Drinfeld double $D_n$ of the Taft algebra $A_n$ defined over an
algebraically closed field $\mathbb k$ of characteristic zero using a primitive
$n$th root of unity $q \in \mathbb k$ for $n$ odd, $n\ge3$, we determine the
ribbon element of $D_n$ explicitly. We use the R-matrix and ribbon element of
$D_n$ to construct an action of the Temperley-Lieb algebra $\mathsf{TL}_k(\xi)$
with $\xi = -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor product
$V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. When $V$ is the
unique self-dual two-dimensional simple module, we develop a diagrammatic
algorithm for computing the $\mathsf{TL}_k(\xi)$-action. We show that this
action on $V^{\otimes k}$ is faithful for arbitrary $k \ge 1$ and that
$\mathsf{TL}_k(\xi)$ is isomorphic to the centralizer algebra
$\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k\le 2n-2$.
