I have a Kenmore 80 series top load washer. It will not go into t
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 (###.#######) and you'll see a list of major sub-components with diagrams and parts.
The Kenmore / Whirlpool top load uses a reversing motor to initiate the pump for draining and the transmission to spin.
If you cannot hear the motor trying to turn the pump, (but the washer will agitate) the problem points to the Timer, or the Lid Switch.
Try to bypass the Lid Switch by removing the Console and jumping the two outermost pins on the switch.
This is a Whirlpool PDF on how these washers operate. It has a diagram on jumping the switch as well as a detailed section on how the transmission works.
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf
If the problem is the Lid Switch, it's a relatively easy fix.
See the following for how to release the console and remove the cabinet.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=131&osCsid=fd38f4b942e68d36aa941a0a4e0c26f7
If the problem isn't in the switch, and the Timer continues to move, AND the washer agitates without any problem, the problem is likely in the transmission.
The problem is likely in the transmission "Neutral Drain Assembly".
Those cams/gears etc. in that assembly keep the washer from spinning until all of the water is pumped out and then go into spin mode.
Ours had the same problem, no spin unless the lid was raised and lowered or we shut off and re-started the cycle.
After looking at the transmission parts list (available on a number of sites) and on the Whirlpool site (Whirlpool makes Kenmore) we noticed that the one of the cams in the neutral assembly which is supposed to have a small spring, didn't on ours when we tore the transmission apart again
http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%20W10114743.pdf
The Sears site for your washer may have a section on the gearcase. Check in that section for a Neutral Drain repair kit.
Found the part number for a kit (Whirlpool part number 388253 Neutral Assembly) for $15.00 and replaced the cams and gears.
In order to get the transmission out, you have to remove the cabinet, the fabric dispenser and both agitators. The top agitator uses a plastic 1/2" drive bolt. You don't have to remove the drum.
http://www.fixya.com/support/r699497-replacing_gearcase
OR ILLUSTRATED DESCRIPTION OF REMOVING CABINET THRU TRANSMISSION.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137
Lay the machine down (after disconnecting power and hoses, remove the pump (two clips), the motor (two screws hold two clips on the motor), lift and set the motor aside, remove two 1/2 inch bolts for the motor mounting plate and three 1/2 inch bolts for the transmission. Pull the transmission, remove the clutch (there is a spring clip that holds the clutch on the main agitator shaft, remove the 8 screws carefully the transmission holds 14 - 15Oz of oil, remove the clip holding the main spin gear on and check the plastic cams.
If the cam that should have a spring doesn't, or the others look damaged, or the main spin gear is damaged, the washer won't go into spin mode.
If that's the problem, install a Neutral drain kit, put everything back together carefully and the problem may be solved.
As an aside, fixing ours cost less than $30 including 80-90 weight gear oil vs. $170 for a new transmission.