The input of the digitizer is required for the iPad mini to boot from Apple logo to full display. Most commonly, the digitizer connector itself gets damaged---you'll see missing gold pins in the connector. Replacement of the connector will be required to bring back the digitizer function which will bring back the display.
I've written a comprehensive blog post detailing a few dozen common pitfalls with iPad mini screen replacements. Check it out. http://www.mendonipadrehab.com/entries/general/the-ultimate-ipad-mini-repair-post