roots
 

How long does it normally take for new roots to grow? Is there anything one can do to stimulate the growth?

Warming the growing conditions, and/or the use of rooting hormones will accelerate the root growth process.

Other than what Ray said, there is no way to consistently determine the time frame it takes for new roots to grow on an orchid. Each individual orchid within a given species or hybrid will perform within it's range of speed of growth for that species or hybrid.