You should be able to cut okra easily with a knife; if there's any resistance, it's too old to cook and should be discarded.
In a shallow dish, whisk together cornmeal, flour, salt, cayenne, and ground pepper; set aside. In a large shallow bowl, whisk together egg whites and 3 tablespoons cold water. Add okra, and toss to coat completely. Lift the okra from egg mixture, letting excess drain off; add to cornmeal mixture, and toss to coat evenly.
In a deep 12-inch skillet or Dutch oven, heat oil to 375 degrees on a deep-fry thermometer. Carefully place half of the okra in the oil. Gently stir without disturbing coating. Cook until golden brown, 3 to 6 minutes; adjusting heat as needed to keep the oil temperature between 300 and 350 degrees. With a slotted spoon, transfer okra to a paper-towel-lined baking sheet to drain.
Return oil to 375 degrees; repeat with remaining okra. Season with salt, and serve with lemon wedges and hot sauce, if desired.