I remember learning in college Russian:

(1) Always use на кухне if you're talking about cooking or eating food "in the kitchen", but

(2) Both на кухне and в кухне are correct if you want to say, for example, "The dog is sleeping in the kitchen (not in the living room)," or "The lightbulb in the kitchen (not in the bathroom) has burned out." (You are speaking of the kitchen as one room out of several rooms in the house, not as a place where food is prepared.)

(3) But when кухня has the abstract meaning "cuisine" or "style of cooking" (not "the room where food is cooked"), в should always be used: Рыбный соус -- очень важный ингредиент в тайской кухне, "Fish sauce is a very important ingredient in Thai cooking." (Here, на isn't possible.)

But I would ask a native speaker if this is correct!