Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?
A clarification: The values $w,r$ are not independent of input combinations: you have already solved for $K/L$.

1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.

2) If there is a profit maximizing pair $(K,L)$, then for all $\alpha \in \mathbb{R}_+$ $(\alpha K,\alpha L)$ will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS). 

A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence $r$ and $w$ will only determine the ratios of $K$ and $L$, not their level.