Step-1: Download official Android 4.4 KRT16S KitKat for Nexus 10 to computer. Rename the downloaded file to update.zip
Step-2: Connect and mount the Nexus 10 USB memory on the computer using USB cable
Step-3: Copy update.zip to the SD card of the tablet. The file should be in the SD card root, not in any folder
Step-4: Power off the Nexus 10. Then enter into Bootloader Mode by pressing and holding Volume Up and Power buttons together until theFastboot menu and Start appears on the screen
Step-5: Follow the navigation instructions on the screen. Press Volume Upbutton for several times until RECOVERY is seen. Press the Power button to select it
source:http://technologygiant.net/how-to-install-official-android-4-4-krt16s-kitkat-update-on-nexus-10/