I've looked at the other thread regarding your scented geranium (Pelargonium) too, but on that one, it looks like mealybug all over the stems. On this one, you say you can see them at the root, but the roots are not visible above ground - I can see (though the image is relatively blurry under magnification) what look like little white insects in your image rather than fungal growth, low down on the stems, above soil level, but I'm not really seeing white cottony masses. These geraniums can get both root mealybugs and mealybug which infests the topgrowth - the treatment is different for both.
To check if there is root mealybug, turn the plant out if its pot and inspect the rootball thoroughly. If you see mealybug in there, follow the directions in this link (scroll down) https://www.joyusgarden.com/fungus-gnats-root-mealybugs/. When you repot, use new potting soil and either a new pot or a cleaned and sterilised (with a scrubbing brush, hot water and soap, rinse thoroughly) pot.
If the mealybug is restricted to the upper parts, above soil level, then, given the lack of success with your previous efforts using soap and water, I'd recommend you seek out an appropriate insecticide that treats for mealybug as well as whitefly, in case it's the latter. Information on Pelargonium pests/diseases here http://www.geraniumsonline.com/pelpests.htm
One other thing - there is a hint of webbing in one or two places on the plant - hard to know if that's from ordinary spiders outdoors or whether its spider mites actually living on the plant, but when you look for an insecticide to spray with, get one that covers spider mite as well as mealybug/white fly.