Broken black plastic on the digitizer connector

Ouch! I know I've made that mistake before, but I was able to continue using it afterwards. The clip should still be able to give tension without the plastic there. A simple piece of kapton tape could help prevent further damage.

I would test the connection by plugging the digitizer and LCD in, and clipping them both down. Power the iPad and test the X,Y coordinates across the digitzer by using a drawing app or simply moving an icon from 1 corner to the opposite corner. If you don't see any skipping you should be okay. If it doesn't have a good grip then the icon will drop at the dead spot, or the line you're drawing will skip. If that is the case then you might have to replace the Logic Board, which isn't cheap! Though you could replace the whole connection, but I personally have no experience in that.