It was successor of ayubid empire founded by Bebarus who fought a decisive war against Mongols at Ain Jaloot. He actually was one who saved Egypt and rest of Africa from Mongols.
Are you sure about these borders? Did the Mamluks never control nubia? Even half of modern-day upper egypt seems missing. The wikipedia page of this sultanate shows a larger map.
I’d love a brief description and timeframe. I’ll go research it, though.