Injury at a laundry mat. Are they liable for medical bills and other expenses?

Although Laundromats are subject to premise liability laws and must follow city and state regulations to provide a safe, hazard free environment, the nature of your injury seems to be self inflicted. It is not clear exactly how this laundry door was defective and it would require further speculation. You can begin filing a claim against the Laundromat as they generally carry liability insurance. If your claim is denied and you feel that it is still the Laundromats responsibility, talk to an attorney and figure out the specifics to determine if legal action can be taken.