Place kidneys with water to cover in a medium bowl; let stand 1 hour. Drain. Remove center cord and excess fat from kidneys; discard. Cut kidneys into small pieces, and set aside.
Roll two-thirds of pastry to 1/8- inch thickness, and fit into a 1- quart casserole.
Combine kidneys, steak, flour, salt, and pepper; toss lightly to coat well. Add onion; mix well.
Spoon steak mixture into pastry-lined casserole; sprinkle with water. Cover with top crust, and slit in several places to allow steam to escape; seal and flute edges.
Place casserole in a shallow baking pan of hot water. Cover pan with greased aluminum foil. Steam pudding at 350° for 3 1/2 hours, replacing water in baking pan as needed.
Note: For a brown crust, remove aluminum foil and pan of water. Place casserole on rack 3 to 4 inches from heating element. Broil 3 minutes or until pastry is golden brown.