Listening to the video one of the officers talks about simply letting the hospital draw the blood, to satisfy the obvious time limit, and then figure out whether the officers can have it later. But this is only after the nurse has been put in the car and things have escalated. To me it seems like this should have been suggested/requested earlier. Who knows if it would have changed the outcome, the nurse clearly didn't understand that in the moment the detective held all the power. She was not going to get out of the handcuffs by trying to tell them what they already knew and citing hospital policy. It's like trying to argue traffic law with an officer that has pulled you over. You just don't do it. You remain silent and argue in court if you want to.