The best first thing to try is to clear/wipe the cache partition....
Clearing the partition will reformat the whole partition, not just delete the files on it. Clearing the cache partition should not delete any stored settings or configuration (but use at your own risk). It is typically done as a troubleshooting step when the tablet is acting strangely, especially with certain apps. My understanding is that the cache is just a pre-compiled version of program files to enable faster launching of apps. If the cache is removed for an app, it will just be rebuilt when the app is next launched. It is not a terribly useful way to gain any storage back, unless you have been running lots of various apps for a long time and no longer use a lot of those still-installed apps. When you delete an app, I believe it will delete the cache for that app at the same time.
http://forums.androidcentral.com/go...451-how-clear-nexus-10-cache.html#post3365755
*Start up the bootloader by holding down volume up, volume down and power on your tablet until you feel it vibe once.
*Use the volume up/down keys to navigate to "Recovery mode", then press the power button. It will appear to reboot.
*When the Android with the red exclamation point appears, hold down power first, followed quickly by volume up. You should now see the recovery menu.
*Choose the wipe cache option. It will take quite a while to format the entire cache partition (several minutes).
*Choose option to reboot.