How Can Thumb Tip Soreness With Flaking Caused By Injury Be Treated?

My wife cut her thumb tip 6 months ago on a bathroom door. Even though it was a small cut, it bleed quite a lot. We washed it and cleansed the area and wrapped with a bandage. Since then, she has bumped it open a couple of times and had to bandage it again. This all started 6 months ago. Now the thumb tip has skin flaking off in the area of the cut and never seems to heal. It looks red and sore. But, my wife says that it doesn't hurt and only itches sometimes. I've looked to see if there is any other indication of spreading infection and could not see any.