This all looks OK.
As you have the "Infiltration" quest, this means that the crystal plug quest is finished. The crystal plug is not in your inventory. but in Nyx', which is as it should be.
Talk to Nyx -> Assign task -> Infiltration
Then she'll tell you to get a map from Gwynn, you get the map, give it to her, again assign the task, equip the plug (in her wardrobe inventory) and send her.