Kenmore 16163 401 dishwasher

Whirlpool dishwashers use a wax motor or a solinoid to release the door at a specific time during the cycle. So for proper cleaning the soap cup is nessesary . The part isn't that expensive and fairly simple repair.

Was the dispenser cup wet when you added detergent?
Make sure the dispenser cup is dry before adding powdered detergent. If dispenser cup is wet, the detergent can clump.
Is the cycle incomplete?
If the previous cycle did not complete, the detergent can become caked in the dispenser cup if it is left sitting in the dishwasher.
Is the detergent old?
Older detergent exposed to air will clump and not dissolve well, which will cause the dispenser door to stick to the detergent. Replace detergent if necessary and store the detergent in a tightly closed container in a cool dry place.
Is the water temperature too low?
For best washing and drying results, water should be 120oF (49o C) as it enters the dishwasher.
Were items blocking the dispenser that kept it from opening?
Items blocking the detergent dispenser will keep it from opening. Make sure water action can reach the dispenser.
Detergent guidelines

• Use automatic dishwashing detergent only.
• Other detergents can produce excessive suds that can overflow out of the dishwasher and reduce washing performance.
• Store tightly closed detergent in a cool, dry place.
• Fresh automatic dishwasher detergent results in better cleaning.
• Add detergents just before starting the cycle.
• The amount of detergent to use depends on the hardness of your water and the type of detergent.
• If you use too little, dishes won't be clean.
• If you use too much in soft water, glassware will etch.
• Suggested fill amounts are for standard powdered detergent. Amounts may vary if you use liquid or concentrated powdered detergent. Follow instructions on the package when using these detergents.
• Water hardness can change over a period of time. Find out your water's hardness by asking your local water department, water softener company, or county extension agent.

Since you mention that your dishwasher has a dispenser has a flap that opens but detergent does not come out, the problem could be the wash water is not getting to it.

Can you confirm if during wash cycle the water is circulating especially through the top wash arm.

Keep a small container on the top rack and see if there is water collected in it at the end of cycle. Alternatively run the dishwasher with door open (you will have to find a way to enable the door switch inside the latch).

Please post a comment to confirm what you observe.

make sure the flap opens and closes freely. If not clean with limescale remover. If it is ok. Your dishwasher may not be filling all the way. Open the door mid cycle and check the water level.

The soap door doesn't open fully on that model, and usually flips fully open as you open the dishwasher door.
It relies on the top spray arm turning to 'wash' the soap out of the drawer - and that will be the problem.The spray arm isn't turning.
Check the spray arm for blocked jets at the very end - use a pin to clear them out, or replace them - check both spray arms. Make sure the spray arm rotates freely also.
Remember the dishwasher isn't a garbage disposal unit - remove food deposits from plates etc before loading the machine. Run a dishwasher cleaner through the machine now and then. Food like pasta, or fruit pips often get carried to the end of spray arms and block them.
If using a 30 min rapid wash program ONLY and dishwasher tablets, put the tab in the cutlery basket as it wont have time to dissolve fully if you put it in the dispenser.

From your description even when the soap dispenser opens it is opening late which is why all the soap does not dissolve.

Try adding only half as much soap and see if it opens then - even a little spilled soap catching on that little latch mechanism can cause it to be difficult to close and stick opening.

Also make sure that the spring that controls the door is not broken.

Dispenser may be bad. Push the hi-temp button , air -dry, hi-temp, air-dry in that order. All lights will lite up and this will do test cycle of all components. If soap disp opens then soap tab may be the issue. I have had intermittent dispenser issues and they are often hard to solve as other components may be at fault.

Use gel run a few cycles and help clear and then don't overfill w gel soap

hi
the dispenser will always open when the washer goes through it's regular cycle. I would recommend to only use powdered soap, that is what the washer was designed to work best with.

both have a wax motor on them that needs to be replaced you just remove the panel and replace the wax motor is what its called