Galaxy Tab 3 Battery Replacement and Screen Repair. Disassemble and fix. - YouTube
What you'll need to do is watch this video (a how to video, you won't replace the battery, ignore the part about the screen replacement, just disconnect and reconnect battery cable), pop the back cover off the tablet, disconnect the battery cable (careful not to damage the circuit board), reconnect the battery cable, and replace the back cover. Connect the charger, within a minute the screen on the tablet will pulse/flicker which is good news, allow it to charge a while, turn on the device, shutdown the wireless wifi, click the power button to turn the screen dark again, and allow to charge for another 6 hours or till fully charged (while it's connected to wifi the background processes will affect the charge time, on a weak charger it may not even charge like from an usb port on a computer). What happened is you allowed the battery to run down till dead, on the newer android OS it seems they want the device powered on while charging instead of allowing the option to power off to charge faster (I think this way they get their metrics from us). It's a pain in the tail, but this will allow you to power back on the device without losing anything or go into developer's mode.