I would suggest you go into ''Settings, About phone'', and tap ''Build Number'' until it tells you "You are a developer." Back out and now you have ''Developer Options'' above ''About phone.'' Enter that, and look for the ''Input'' section. Enable ''Show touches'' and ''Pointer location''. Now touch on your screen, and "draw" down to the soft home keys.
Does the screen indicate registering your screen touches over that area? If it does, then the screen is working, and there must be something else wrong. Depending upon your ROM and version, you might need to factory reset your phone, or flash a [https://developers.google.com/android/nexus/images|factory image from Google].
Disable ''Developer options'' before pursuing other avenues.