This was recommended to me by a periodontist I was referred to a few years ago for something similar but not so drastic.
Buy a Water Pik and some Betadine. Every night, fill the Pik tub with warm water and a teaspoonful of Betadine. With the pressure setting at 1/2 to 3/4 full power, begin to irrigate the area at the gum that's the problem. It will be very tender at first, but keep it up every night and go ahead and do all your teeth at the gumline inside and out. I had something similar and it cleared up in a week. But you still have to do this at least every other night. And every OTHER night, consider brushing your teeth with a pasty mixture of baking soda and hydrogen peroxide, again, his recommendation. This has been my regimen for years.