Link to the photo of the plant

Here is a link that might be useful:

Mealybugs, all of 'em. Adults, ovasacs, crawlers, etc. That's a very bad infestation of a pest that can be quite a challenge to control.

Have you tried anything, so far? I would ordinarily recommend a misting solution of water/alcohol, but I have never treated coleus with the alcohol.

If this were my plant, I would take as many small stem cuttings as I could, wash them all carefully in soapy water, rinse, wash, rinse....and then root them in a good potting mix. Do you know how to take stem cuttings?

I have yet to see a Mealy Bug that can take a good coating of Horticultural Oil and survive. Even at low concentrations of 1%. I do not know if Coleus is sensitive though. I think the reason people don't always get control with Oil is that they spritz it on like a stomach poison pesticide, Oil works only if you completely coat the insect.
You probably don't have time for this but the Mealy Bug Destroyer, Cryptolaemus montrouzieri is a predator that is commercially available. Predators take the right conditions and pest populations to establish, but this one seems to be a little more forgiving than some others. It will also eat other pests occasionally. Besides they're pretty cool bugs.

Here is a link that might be useful: Mealy bug destroyer

Like Aphids Mealy Bugs can be knocked off any plant they are on with a shapr water spray. Coleus is sensitive to horticultural oils so that may not be a good option.

I disagree that mealies can be sprayed off with water, at least not enough to manage a population consisting of generations of the little pests.

I have Coleus listed on the actual Oil label listed as a safe plant. Of course always check with a test sample. Also under certain conditions Oil will burn any plant, so read the label.