The police cannot bust down someone's door for refusing to talk to them. Without a warrant, the police can only enter a private residence either at the invitation of the owner/occupant or if there are exigent circumstances, such as evidence that someone is in danger or in pursuit of a fleeing suspect.
The police are incorrect in suggesting that Dana would have to come with them to the hospital. The police cannot compel someone to receive medical care against their will, even if the person is injured.