You are having a bacterial infection on the lips and as they have become hard, you will need a dual therapy.
I would suggest starting with neosporin
antibacterial cream apply twice daily and after half an hour, apply 0.5% or 1% hydrocortisoe cream only to the affected area, NOT inside the mouth or anywhere else.
Continue this for maximum 10 days, and if you see an improvement within a week, you can stop.
Please see a doctor if there is no improvement, as you might need an anti-fungal as well.