You don't need to replace the LCD screen, you've accidentally blown the iPad mini backlight fuse by inadvertently removing the LCD connector before removing the battery connector. You will need to have the iPad mini backlight fuse replaced on the motherboard---other solutions like jumping the backlight fuse are possible, but are likely to eventually overload the backlight ic and cause it and/or the backlight coil to fail.
For more than you ever wanted to know about iPad mini backlight problems check out this thread.
Lots of advice here on this thread: http://www.ifixit.com/Answers/View/126805/Screen+stays+black+after+digitizer+replacement