First, do not drop multiple items or stacks, only 1 item at a time or the excess is disgarded.
Second, if "Hawk Girl says nothing after dropping the item, then you did it correctly.
Third, to perform this "trade," simply drop the acceptable item into the Hawk Girl's" nest. Then quit your game via the menus. Simply reload and your dropped item will have converted!
Here is the list:
Bloodred Moss Clump ---> Twinkling Titanite Blooming Purple Moss Clump ---> Twinkling Titanite x2 Cracked Red Eye Orb ---> Purging Stone x2 Dried Finger ---> Blue Titanite Chunk Dung Pie ---> Demon Titanite Egg Vermifuge ---> Dragon Scale Humanity ---> Ring of Sacrifice Pendant ---> Souvenir or Reprisal Prism Stone ---> Demon Titanite Pyromancy Flame ---> Red Titanite Chunk Pyromancy Flame ---> Red Titanite Slab Purple Moss Clump ---> Twinkling Titanite Rubbish ---> Titanite Chunk Ring of the Sun Princess ---> Divine Blessing x2 Sack ---> Demon's Great Hammer Skull Lantern ---> Ring of Fog Sunlight Maggot ---> Old Witch's Ring Sunlight Medal ---> White Titanite Chunk Twin Humanities ---> Rare Ring of Sacrifice Xanthous Crown ---> Ring of Favor and Protection
Here's a question- I understand that you can't drop more than one item at a time, nor can you drop one item, get a trade, and then trade for the exact same item again. However if I, say, dropped a Bloodred Moss Clump in exchange for a piece of Twinkling Titanite, would I be able to leave the Asylum, come back, and the do another tade for another piece of Twinkling Titanite?