While she is correct that it is illegal for you to have access to another patient's information without their consent ... The instant they hooked your father up to that IV, it became part of his medical record. So if you have permission to access his medical information, you have every right to take the picture and keep it.
I would also suggest you immediately contact the hospital's "Quality Control" department or "Administration", and ask them what they are going to do about your dad being given someone else's IV.
As to recovering the photo ... It depends on your phone and carrier. I would suggest contacting them. But if you don't automatically upload your photos, they're probably deleted for good.