Charge your battery with a [http://www.witrigs.com/extra-battery-charger-adapter-kit-for-samsung-galaxy-s3-black|charging box] to make sure if your battery okay or not.
If the battery is okay, you should try to turn on the phone again with full power battery.
If still not work, you should make sure all of the connectors to logic board are connected very well and are clean without much dust.
If still failed, exchange your display with another working well note 2. On the one hand to ensure your new screen is good or not on the other good mother board. On the other hand, make sure your mother board is good or not with the other good display.
Good luck to you!