Chosen Solution
iPad mini doesn’t turn on after screen was changed.
Was the battery disconnected when disconnecting and connecting the LCD display cable? If the battery was left plugged in the chances are the backlight filter blew up which will require someone with micro-soldering experience to replace said filter. You can confirm if filter is blown if you can make out dim image display by shining a flashlight behind or on the screen.
Try connecting it to a wall charger the battery may be dead. Try disconnecting and reconnecting the ribbon cables of the screen. Are you sure your screen is from a good quality source.