Can I fix this problem with my dishwasher?

close the flap turn the dial thru the cycles, to see if it will click, flush out the box with water, some times a soap build up will cause it to stick,

You stated that the water pipes were frozen. Was the whole house unheated ,did all the pipes freeze? You should not hear the motor running unless there is water in the dishwasher. Your dishwasher should be connected to a hot water inlet. A solenoid valve opens up and fills the tub with hot water. Then the float senses the water and the dishwasher starts its cycle. Shut the valve off to the dishwasher, undo the connection and turn the valve back on 1/4 of the way. Make sure you have a water supply. No water , no washy.

It sounds like the solenoid or other actuator that unlatches the detergent dispenser has gone bad in some way.

If you or someone you know is handy with simple repairs, it might be worth pulling the cover off the front door (assuming the dispenser is on the inside of the door) and surveying the actuator. If it looks intact and doesn't have corrosion or any obvious defect (broken wire, burned coil, etc), then the problem might be in the controller itself, and that's probably going to be fairly expensive to fix.

Try just tossing the soap in the bottom of the dishwasher, since some have the cup without the flap it must be okay.

get your model and serial then go to Maytag web site and check dishwasher recall to your model # if its under the recall Maytag will replaced the whole door

Not worth the effort..Fork over the bucks or wash by hand.

try those throw in cubes to wash with and bypass the dispenser.