Look at the back of the dryer at the very bottom you will see a black bung.
Take this off and let the water drain out so you will need a towel for this.
Once the water has drained out refit the bung and retry the dryer and see if it starts going then and let me know.
Ok so if you keep the start button held in will the dryer start running untill you let go?
Ok im with you now.
The problem will be the switch itself. To cure this you will need to take the top off and from there you will see the switch.
From there you will need to replace the switch and then the dryer will be back to normal then.
Click here for the part
To replace it you sometimes need to remove the front control panel so its up to you weather to fix yourself.
But as its a landlord dryer you will proberly be best to get them in to fix it so theres no risk for you.