I have a Kenmore (Whirlpool) HE Top loading washer model 1102613412. It works okay but it clicks during agitation when it gets to the end of the "swish" range.

SOURCE: washer will not agitate

A wears out clutch might prevent the washer from agitating well or at all. If the clutch is worn, you need to replace it. Good luck!

SOURCE: Kenmore top loader Model 110.20712990

It could be the coupler, but I doubt it. More commonly, this is the symptom of a gearcase (transmission) going bad. Couplers typically just snap off or the center hole rounds out causing it to slip. The result: the washer will fill and drain, but will not agitate or spin. The telltale sign of a bad gearcase is the grinding noise you mentioned and the requirement for you to assist the spin under a load. In addition, if you inspect under the washer, sometimes the gearcase will begin to leak a honey colored oil. You may find droplets under the washer, and in extreme cases, you may find spray patterns along the interior cabinet walls.

Now before the sense of panic sets in, replacing a gearcase on this model washer is not that difficult and it can be accomplished by the average do-it-yourselfer with some guidance. I have a link that explains step-by-step how to replace one:

http://www.fixya.com/support/r385250-replacing_motor_coupling_kenmore_top

Read through the link thoroughly and see if this is job that sounds like you wish to attempt on your own. No special tools are required. A screwdriver set and socket set is all you should need.

A replacement gearcase can be found at searspartsdirect.com. Just type in your model number and locate item number 9 under the "Brake/Clutch/Gearcase/Motor/Pump" heading. The part number is 3360629. If you are going to replace the gearcase, it is also recommended that you replace the motor coupling as well. The coupling is listed under the same heading as item number 13 (part number 285753A). You can also check out repairclinic.com to see if they carry the parts any cheaper. Use the part numbers provided. I will post the link that explains how to replace the coupling as well:

http://www.fixya.com/support/r385250-replacing_motor_coupling_kenmore_top

Many of the steps are a repeat, but I try to be thorough. Some of the original motor couplings that came from the manufacturer would not last long under constant use, because they were made entirely of plastic. The manufacturer modified this part to include a steel grommet that makes it much more durable. The new part is listed with the part on the website, and included in the link I provided.

NOTE: If you do order a replacement gearcase make sure you follow the instructions on the outside of the box. There should arrows indicating that the gearcase needs to be shiped in the UPRIGHT position. If not, transmission oil will leak.

I hope you find this information helpful. The staff asks us to refrain from direct contact with the consumer and only allows us to answer questions in this forum. I hope you don't mind. Let me know if you require additional assistance.

SOURCE: Kenmore 80 Series Top Load Washer not agitating

Check out this you tube. Had the same problem today. Tipped washer on its side and replaced coupler between motor and gearbox. Parts are about $15
This is a good diagnosis video.
http://www.youtube.com/watch?v=VYOP0_KCYT0

SOURCE: kenmore elite won't agitate properly

Does the bottom of your agitator work correctly? We had the same problem if you pop the cap off the top of your agitator there is a cam in there with 3 dogs that engage the agitator causing it to spin. If the bottom part is agitating correctly then chances are the dogs in the cam assembly are shot. If the bottom is agitating correctly then it is very possible you have something stuck under the bottom section of the agitator. If this is the case remove the top cover (same as before it's just a lid that pops off) inside you will see a 1/2" square use a 1/2" drive ratchet with a 3" or 6" extension unscrew this piece remove the top section of the agitator then the bottom. Once the bottom is out check to make sure all is clear inside the washer. Reassemble in the reverse order making sure the lower section sits all the way down on the splines for the drive mechanism. If neither one of those solve the problem I would suspect you have an issue in the drive train of the washer.

SOURCE: kemore elite top loader won't fully agitate

If you have a whirlpool built machine th problem is probably the agitator dogs. They are located inside the agitator. My guess since you say it doesn't fully agitate. If changing the dogs does bot fix the problem then the coupler is probably bad and that connects the motor to the agitator.