Lugol's simply isn't strong enough. Depending upon the severity, I'd use and have used 7% and 12% tincture. The latter just works way faster. Obtaining the 12% outside of making it yourself is a bit trickier.

It's been a long time since I researched all things iodine, so I can't cite sources, but in my experience it absolutely (somehow) encourages tissue to regrow as it originally was. Not only was I able to remove a decades old facial scar from a motorcycle accident (mainly from the stiches) using topical tincture, after taking Lugol's internally over a short period I had feeling returned to the top of my foot which had been numb from an accident 30+ years ago. I absolutely credit the nerve regrowth to the iodine.

I guess I should also mention that the scar was to my right eyebrow area and that hair grew back where it hadn't been in decades.

Other than that, I've heard reports of stretch marks and even caesarean section scars eliminated/lessened from various oral proteolytic enzymes such as serrapeptase, et. al.

A couple of drops of SSKI as a topical will take care of infection, even if it lies well beneath the surface.
If I understand correctly from your post it hasn't fully emerged, I can relate my experience when a poor college student. I had a wisdom tooth that was begining to emerge and kept getting infected when I'd accidentally bite with it during eating. Dentists' offices rejected my inquiries on having it simply incised so that food wouldn't get trapped under the gum. It got badly infected to where it was 'numb' with pain. So, I took a double edged razor blade, broke it, sterilized it, and sliced along the crown of the tooth. I then packed it with salt several times a day using a broken toothpick. The salt ate away the excess gum and once exposed, the problem was rectified.
Not saying you should do this, especially if it is coming in at an angle, but salt is a way to both lower infection and remove gum. The SSKI will definitely heal gum disease in general to the point of closing pockets within a very narrow time frame.