= Solution
Take an integer $k\geq0$, so $k\rho$ is a <dominant integral weight>. Apply the dilation $e^\mu\mapsto e^{(k+1)\mu}$ to the <Weyl denominator formula>:
$$
\sum_{w\in W}\det(w)e^{(k+1)w\rho}
=e^{(k+1)\rho}\prod_{\alpha\in R^+}(1-e^{-(k+1)\alpha}).
$$
This is exactly the numerator of the <Weyl character formula> for $L(k\rho)$. Divide by its denominator and use a finite <geometric series> in each root direction:
$$
\boxed{\operatorname{ch}L(k\rho)
=e^{k\rho}\prod_{\alpha\in R^+}
\frac{1-e^{-(k+1)\alpha}}{1-e^{-\alpha}}
=e^{k\rho}\prod_{\alpha\in R^+}(1+e^{-\alpha}+\cdots+e^{-k\alpha}).}
$$
This <character of a Weyl-vector multiple> also gives $\dim L(k\rho)=(k+1)^{|R^+|}$ by evaluating every formal exponential at one. This evaluation is safe in the final polynomial expression, without dividing by a zero denominator.
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