Kenmore Fridge - won't stop making ice

check the tubing for ice in the tubing at the back of the fridge.
If Ice, thaw it out by putting in hot water then reinstall it in fridge and make ice.

I have not worked on this model before but on the whirlpool they have a bar as well. you can turn off ice maker by pushing it up and it locks in place to stop ice production. if u want it to make ice u push it up slightly and it should move down. also check the ice maker tray to be sure u are getting water to the tray. if u get the bar to go down and no water comes in then check that u have water at the back of fridge.. if u have water there but not in the tray then you may have a bad water inlet valve on back of fridge.

Does your ice maker have the on off switch on the front of it?

seal on door must be letting moisture into compartment, need new door seal

Ice in the path of air flow to fridge. The freezer also kkeps the fridge cold via a fan that blows the cold air from feezer to fridge.

Ice makers do go bad and it is inevitable. Check the ice maker connection and make sure that all is in order. If all seems okay and the ice maker does not work, replace it.

5 reasons for fridge and/or freezer getting warm: 1- the evaporator coils are icing up and won?t let air circulate. 2- the evaporator fan is not working to circulate air. 3- there is no refrigerant in the system. 4- no power to fridge or compressor. 5- the condenser coils have no air circulation check under the fridge behind the vent. if it is too dusty or the fan isn?t working, it is not getting enough airflow to cool down. replace fan and/or vacuum out the dust. if there is no power to the fridge, the light will be off. if the compressor is not running, there is a defrost timer that cuts the power temporarily to the compressor to melt ice on the coils. it?s located under the fridge behind the bottom vent. If the timer is off and the compressor is not going, the compressor or the main board is bad and needs replacing. there is also a temperature sensor on the evaporator coils that tells the defrost coil to turn on and when to turn off. if the defrost timer is on but the coil is not heating up, replace the sensor. if it still won?t heat up, it?s the timer or the coil. the easiest way to tell if there is no refrigerant- the freezer wall is warm and the compressor is loud. if there is no refrigerant, it may be more cost effective to replace the fridge. if you feel no air coming from the freezer vent, the fan needs replacing. If you feel air moving in the freezer but not the fridge, the channel from freezer to fridge is blocked. if the freezer has air and the rear freezer wall has frost on it, the evaporator coils are iced up and need defrosting. causes: 1. low refrigerant 2. no fan 3. too much humid air in fridge/freezer due to door open or bad door seals 4. the condensate drain tube is clogged if the seal around the door is ripped or not completely sealing, or there is condensation at the bottom or the walls of the fridge, replace the seal(s). no air, no fan. Replace if the refrigerant is low, it just needs a top off. but you need to be certified by the epa to move refrigerant, so get a professional. defrost solutions: unplug the fridge. remove the wall of the freezer that is frosted/icing up. use a hair dryer to melt the ice. do not get too close. you will see a drain under the evaporator coil. make sure it is not blocked. the water should run freely to a condensate pan under the fridge that evaporates the condensation with the hot line from the compressor. once the coil is free of ice, frost, and water, replace the panel and repair any of the problems that caused the icing up. plug the fridge in and turn it on. If it frosts up within a week, there is still an issue that not been taken care of.

You might check out this site and ask a question. They have drawings on many of the fridges and I have bought many parts from them.
http://www.appliancepartspros.com/partsearch/

I would check that fan motor in the freezer section is running. it distributes cold air from freezer into refrig section also if this motor is running the top portion of freezer will not get cold enough to allow ice production. the motor is easy to repl and is mod# specific you should be able to buy one online.

sounds like your freezer needs repair.