If this is a Kenmore Top Loader washing machine, it sounds like you have a broken motor coupling. The washer filling and draining, but no agitation and spin are common symptoms with this problem. There may also be a rattling or whirring sound as the drive motor attempts to engage.
NOTE: If the washer drains, the drive motor is working. This drain pump on this washer is driven by the drive motor.
The motor coupling is located between the back side of the drive motor and the gearcase (transmission). The coupling is used in place of a drive belt, which is why this type of washer is referred to as a "direct drive". The following link explains how to access and replace the motor coupling:
The part number is included with the link and can be purchased at appliancepartspros.com for about $11. I have included an illustration of what the coupler looks like below.
If the motor coupling is intact upon inspection, the other source of no agitation and spin would be the gearcase. The gearcase on these washers is fairly simple to replace. If you narrow it down to a failed gearcase, the next link explains the steps for replacing one:
I could only find your model number referenced at repairclinic.com. The gearcase on their site lists for about $160 and is listed as item 423.
I could not find any exploded view diagrams of your washer, but the majority of all Kenmore and Whirlpool top loaders are designed with the same direct drive system. You can purchase a Direct-Drive Repair Manual at repairclinic.com for about $20 (Item number 1170635) if you need a guide for reference.
If you have any questions about this repair, or require additional assistance, please let me know. I hope I've given you enough to work with in trying to narrow down the cause of your problem.