Washer hums when turned on will fill with water

This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.

First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number (110.#######) and you'll see a list of major sub-components with diagrams and parts.

Kenmore / Whirlpool top load, direct drive washers use a "Motor Coupler" between the motor and transmission.

This is the most likely problem.

It's used as a fail safe device and consists of three parts.

Two of the parts are the same, plastic units with three prongs that fit on one of the motor output shafts and the other on the transmission input shaft. Between them is a rubber piece with six holes.

The power from the motor is transmitted through the rubber piece to the transmission.

The other shaft on the motor directly drives the pump.

If the motor coupler is broken, or worn, the motor may not be able to send full power to the transmission for agitating or spinning.

See the following for how to remove the cabinet, pump, motor and motor mounting plate to inspect and replace the motor coupler.

http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137

If the coupler looks damaged, it's a relatively easy fix.

Pry the plastic pieces from the motor and transmission. Some people use a claw hammer but I've found that using a small (6 inch) pry bar works well.

Carefully fit the plastic pieces over the motor and transmission shafts and press them in place (Use a 1/2" of larger socket to

put pressure equally around the pieces.

Put the rubber piece on the transmission plastic piece.

Replace the motor mounting plate.

Carefully line up the plastic prongs on the motor based plastic piece to the three un-used holes in the rubber piece. If they don't line up right, the coupler could be damaged.

Mount the motor, don't forget the two screws, put the pump back on the motor and use its' clips.

Put the cabinet back on, don't forget to plug the lid switch back in and re-mount the console.