You write them a letter stating that the condition was caused by the previous tenant. You tell them that the screen is useless and you cannot leave the door open without insects coming inside. You tell them that your verbal requests have been ignored and you are making a formal written request. You give them a reasonable time to replace the SCREEN - the DOOR does not need to be replaced. 10 days is reasonable.
Then you mail the letter in a way that provides you with proof of delivery or attempted delivery. USPS certified mail is the best way.
Then you watch the tracking. Start counting the ten days from the date the letter was delivered, or from the first attempt. Sometimes letters will be ignored and in court, ignored letters do not protect the person being sued.
If it's not fixed, follow the same procedure with a second letter, that says at the top 'Second Request'. Give 10 more days.
If this still doesn't get it fixed, send a third letter that says 'Final Demand'. In this letter, you state that if the work isn't done in 10 days (it's now more than 30 days at this point), then you will follow your state landlord/tenant law in either replacing the screen yourself and withholding the cost from the next month's rent, or if the state doesn't allow that, you will sue in court to have a judge order the screen to be replaced and pay your court costs.
Sometimes all it takes is the first letter to make a landlord realize you are serious. Sometimes it takes going all the way.