Allowing the detergent to soak for several minutes prior to scrubbing to loosen the contaminants is a good idea. I'd also suggest vacuuming beforehand with a new or nearly new filter bag (or an empty cloth bag) to effectively prepare the surface. Another consideration is that multiple scrubs may be necessary to yield the best results.
In addition, manufacturers of electric rotary polisher-scrubbers (e.g. single brush -types such as Johnson Wax or Sunbeam, double brush-types such as Hoover & G-E and triple-types such as Aerus & TriStar) recommend(ed) using the scrubbing brushes for usual cleaning and attaching scouring pads to the brushes for aggressive cleaning in sections such as 3- by 6-feet.
Kirby's roller brush-type polisher attachment is a different option which would be effective, too.
You might also apply a bleach solution after scrubbing to brighten lighter-colored grout may .
Finally, adding white vinegar or apple cider vinegar to rinse water often helps remove any soap residue.
Best regards. Keep us posted.