Other mistake: At the end, before House's "realisation" the camera pans out and shows a street light when House and Wilson are outside talking. The lamp top is covered with fresh snow. It's been dark for sometime. The heat from the bulb would have melted snow had it actually been real.
Other mistake: When Kate asks House about his insomnia, he reacts by turning to look at her, except that he is now holding the computer in front of him and should be looking there to look at her, not at the coffee table. (00:17:46)
Answer: She would have No Reason to know it belonged to the CIA. If she did know he went off in the helicopter, all she would know is that it wasn't an ambulance helicopter.
Greg Dwyer