Yup. Cafe Nero right at the door leading to the Staff Travel office. Departures level, north end of the building - Zone A.
About four or five internet kiosks.
If I were you, though, I'd head through the door leading to the staff travel office and when you spot the office on your LEFT you'll see a door opposite marked "RED DESK/BLUE DESK" or similar. Head in there and speak to one of the staff and ask them to list you on the flight. It'll be easier and they'll be able to give you an indication as to availability.