Continuity mistake: When Ray Winstone takes his pill with whiskey while talking to Mel Gibson, he takes a swig of the drink and knocks the pill back, leaving some alcohol in the glass. When we cut to a wide shot his glass is completely empty.
Continuity mistake: In the scene after Craven talks to the Senator, the vehicles beside his police vehicle change from the shot showing him enter the car to the next angle showing his car leave its spot.
Chosen answer: Possibly from getting his daughter's blood on him, when he accepted a drink from the Northmoor executive, or from handling his daughter's radioactive belongings. Given the depth of the coverup, they may have even anticipated that she'd come to him and broken into his house and poisoned him before the story even began.
Captain Defenestrator
The father and daughter died of air exposure. We know the three people died instantly via radiation steam. If the daughter ingested the radiation, then the radiation would only end up as the hair grew. In this case, the length of hair was very long, so she had to have ingested the radiation for a long time. Not to mention her hair would have fallen off. Since her hair was healthy and radiation detected whole hair, she was exposed to radioactive dust. The milk was there for backup, and it was full.