Here's what I did when my old WAL got an infection of some sort.
I dried off the affected area plus some of the area around it.
I removed the fuzz (dead skin, scales, etc.) without taking pieces of the lungfish with it.
I applied hydrogen peroxide solution (standard 3% solution) to the affected area and the surrounding area.
I rinsed the lungfish with clean water.
I applied Betadine to the affected area and the surrounding area.
I rinsed the lungfish with clean water.
I applied a large gob of Neosporin to the affected area.
I placed the lungfish in its hospital aquarium that contained clean water with 1 tbsp salt per 10 gallons of water. The aquarium was kept at 82° F until the infection cleared.
I repeated all of the steps above as necessary on a daily basis.
I did 50% water changes every other day, and I made sure that the salt levels were kept at 1 tbsp per 10 gallons of water.
The infection on my lungfish was fairly large (size of a quarter) but not as large as the area on your lungfish; it took about a week or two to go away. The scales on the lungfish in that area were a bit messed up for a few weeks after that because of the infection, but those went back to normal, too.
Try that procedure that I listed and update us on the lungfish's progress. I'm going to be honest with you, though: I can't say for sure if it will make it because of how large the affected area is and how damaged the back part of its fin is, but lungfish are extremely tough, so it may still pull through.