1. Place potatoes in a medium saucepan; cover with water. Bring to a boil. Reduce heat, and simmer 8 minutes or until just tender; drain well. Cool to room temperature in colander.
2. Place potatoes in a large bowl. Add 2 tablespoons flour, salt, and garlic; mash with a fork or potato masher until smooth. Cover and chill 20 minutes.
3. Combine cheese, corn, and jalapeños.
4. Place 1/2 cup flour in a shallow dish.
5. Divide potato mixture into 16 equal portions. Flatten each portion into a 2-inch round patty. Working with one potato patty at a time, top with about 1 1/2 tablespoons cheese mixture. Top with 1 potato patty; press gently to seal edges. Carefully dredge in 1/2 cup flour, brushing off excess. Place potato cake on a plate. Repeat procedure with remaining potato patties, cheese mixture, and flour to form 8 potato cakes.
6. Place egg whites in a shallow dish. Place panko in another shallow dish. Working with one potato cake at a time, dip potato cake in egg whites, allowing excess egg white to drip off. Coat potato cake completely with panko, pressing gently to adhere. Set aside. Repeat procedure with remaining potato cakes, egg whites, and panko.
7. Heat a large nonstick skillet over medium-high heat. Add butter to pan, swirling until butter melts. Add potato cakes to pan; reduce heat to medium, and cook 3 minutes. Carefully turn cakes over; cook 3 minutes or until browned.