Close the valve in the supply hose attached to the Kenmore. The valve has a round knob that twists clockwise to stop water from reaching the refrigerator.
Disconnect the Kenmore power cord from the wall. Pull the refrigerator far enough from the wall so you can crouch behind the appliance to inspect the back plate.
Remove the bolts in the water service panel where the supply hose enters the Kenmore, using the nut driver.
Pull off the service panel. The water valve and solenoid assembly are immediately behind the panel. The solenoid is on top, with two wires clipped onto the conductor blades.
Pull the wires off the conductor blades, gripping each wire by the metal fastener on the end.
Dial the multi-tester selector control to X1, which tests for electrical continuity. Place the rods on the multitester on the two conductors on the solenoid. A normal reading falls in the range of 200 to 500 on the display. A reading outside this range, or no display at all, means the solenoid has failed. Buy a replacement part at Sears, which sells Kenmore refrigerators, or an appliance repair shop.