 Solution

Take an integer $k\ge 0$, so $k\rho$ is a dominant integral weight. Apply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the Weyl denominator formula: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: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.