Emancipation - S5-E8
Continuity mistake: At the very beginning, House walks in and puts his cane on the table on top of Thirteen's papers. She picks it up and moves it away, placing the bottom end on a stack of books. Cut to House and the cane is now flat on the table. (00:02:55)
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