As far as I am aware, below are the Libraries availble:
Dwarfholme University's Library in the Brotherly Cities, the Barony of Dwarfholme, the County of Dwarfholme, the Duchy of Eastland.
Ravenswood Public Library in the city of Ravenswood, the Barony of Ravenswood, the County of Brunston, the Duchy of Eastland.
The Old Capital Library in the Old Capitol City, the Barony of Koroshi, the County of Augustus, the Duchy of Eastland.
The Library at Three Wells, in the City of Three Wells, in the Barony of Chesterdell, the County of Brunston, the Duchy of Eastland
Since we just stayed in Asyl, the Library at Three Wells would be the closes Library to maximize your study time. From there, the Ravenswood Public Library might not be a bad choice. The Old Capital Library is near Valbourgh, which is essentially across the whole of the Eastlands. Im sure Banrotti or Al would be able to give you the Wayfarer code to Valbourgh so you could get to this Library quicker.
Unfortunately, I do no know the detailed information that each one of these Libraries specializes in. For your idea of mixing elemental essences with crafting, I would either start with Ravenswood or the Library at Three Wells and see what they have to say. Maybe the Librarians there could even point you in the direction of other Libraries that specialize in that.
Good Luck.
Finley
PS - No need to pay; Helping people is what the Vitrium Ordinance is about!