Coming around 3,000th most common of the words in Project Gutenberg, we begin close to the heart of Bag End, but Gollum knocks all thoughts of kitchens out of the story.  “Kitchen” is a word of Dutch and German heritage, but ultimately from the Latin verb coquere, to cook.

01.002 kitchens, dining rooms
01.059 and Dwalin at the door of the kitchen,
01.068 in the middle of the kitchen trying to see what they were doing.
02.001 in the kitchen.
02.002 in the kitchen before turning out the dining-room.
05.002 in his own kitchen at home –

“kitchen, n.” OED Online. Oxford University Press, March 2016. Web. 10 March 2016.