When applying vaseline or aquaphor, make sure your lips are always moisturized and never let them dry out. That should stop the buildup of dead skin. If you see any changes like new skin developing (you'll notice it by the grayish color) then your lips are healing. Then again, there's a chance it might not work for you. It all depends on how damaged the dermis is.
If aquaphor or Vaseline fails, then I recommend you purchase Gloves in a bottle. From what I can see, your lips don't really look that bad, so Gloves in a bottle is very likely to heal you. (Mine was just as bad as Daniel Miller's and it worked for me). This product healed that white line you mentioned.