See the attachment for the tech sheet.

I am not sure why you think it is a defrost problem.
Did you see that the evaporator/freezer coils where iced/frosted over to the point that the evaporator fan cannot pull air through them?

Also if it was a defrost problem after you did the manual defrost the compressor should have come on and stayed on till the freezer reached set point temperature. Then it should have cycled normally until the frost built up to the point where it clogged the evaporator coils. This would probably take a few days. It then would come on and stay on all the time.

It would be a good idea to measure the actual freezer temperature with a thermometer. It should be 0 to 5 degrees F. If lower you may have a temperature control problem.

Is the condenser/compressor fan also running all the time? It should be on whenever the compressor is on.

Here are your parts
Parts for Kenmore Refrigerator 10656693501 - AppliancePartsPros.com

If you go to the Sears parts site they have actual part breakdown diagrams that may be useful to you.Unfortunately they do not share these with other sellers.

First remove the evaporator cover in the freezer so you can see the coils.
If they are heavily iced/frosted over then it is a defrost problem.
Do not let them de-ice.

If yes.
Manually force a defrost cycle, see the tech sheet I attached earlier. When the fans and compressor turn off you are in defrost.

Now check the defrost heater to see if it is on.
Be careful you do not want to burn your fingers.
If the heater is on then the control board needs replacing as it is not programming in the defrost cycles.

If not on.
Unplug the unit.
Remove the wire for one side of the heating element from the wiring and measure it for continuity, usually around 20 ohms or so.

If the heater is OK
Remove one wire to the defrost thermostat and measure it, should be 0 ohms when frozen. Note that it opens just above freezing so must be frozen to check it. Also inspect it, if it is bulged at all replace it even if it measures OK.

If both the above are OK then odds are the control board is toast.
Only other thing would be a bad connection from the board to the defrost heater/thermostat combo.

Take a look at the tech sheet.
Go into the service mode and it is test #6

[COLOR="Blue"]There are wires of red, brown, green , white and black colors. Of what wires exactly should I measure the voltage?[/COLOR]
What voltage are you referring to?
[COLOR="Blue"]
What is the black unit with red/brown wires and next to the sticker "556"?[/COLOR]
That is the defrost thermostat.

Should be Pink to White according to the wiring diagram.

One side of the defrost thermostat goes to the control board (pink)
The other side of the thermostat goes to one side of the heating element (brown).
The other side of the heating element goes to Neutral (white)