First of all, it's a battery operated device and what you are plugging in is the charger. Though they are usually shipped partially charged yours may have been depleted for some reason. Plug it in and leave it alone for a few hours and come back and try again. If you have done this already then try a soft reset by holding the power button down and not releasing it until approximately 30 seconds have passed. After releasing press it again to see if it comes on. If it still fails to turn on I would contact Amazon support and let them know that you may have a defective unit. They might have you try a few other things before calling it defective but since it's brand new and under warranty if it is found to be defective Amazon will replace it.
At the very least you'll need a LED/LCD panel tester. They usually run under $100. Of course you will also need the correct connector for the device you are testing. Most kits come with standard cables but you will need to purchase specific ones for devices such as phones and tablets. Here is an example of one on eBay - http://www.ebay.com/itm/LED-LCD-Panel-Tester-For-TV-Laptop-Repair-Support-7-84-LED-LCD-Screen-/272265411197