Your model number is missing digits at the end of it.
Kenmore numbers are 3 numbers, a period and then 6 to 8 numbers.

In most units the fill cycle is just controlled by a time.
There is nothing in the unit to tell it to start once the unit is full.

Most units fill to just under the heating element.

Often if the fill is low you will also hear a growling sound because the pump will cavitate (suck in air).
The most common cause is a dirty screen filter on the input side of the fill valve. Remove the valve from the unit and clean/check the filter.
Do not permanently remove it from the valve as it is there to keep crud out of the valve which could hold it open and cause a flood.

If it still does not fill correctly I would replace the fill valve.

PS: Be sure to shut off the power and the water supply to the unit. Have paper towels etc. handy to wipe up any spilled water. Get a small pan/bowl etc. that will fit under the valve then loosen the input water line let it drip into the container emptying it as necessary. This will minimize the amount of water spilled.