Sounds like a bad digitizer. Of all the iPhone and iPad replacement parts out there, the most problematic parts are iPad mini 3 replacement home buttons, iPad mini/Air dock ports and then iPad mini digitizers.[br]
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If you long press the top power button after the device boots, as if you were going to slide to power off, this should prompt the display to come up. Hit cancel and then use as normal. The iPad mini smart cover hall effect sensor and therefore power button is routed through the iPad mini digitizer/home button connection. The hall effect sensor lives on the new screen and is often missing/damaged on crappy replacement screens. Crappy replacement screens are the common screens easily available, it is really hard to find a gold bordered original quality mini screen. [br]
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If you don’t have display come up after long press then something else is up and you can post back.