Once you have 44 Uni 34s, you can begin "Nulgath Demands Work" from Oblivion. Complete it 4 times. The Unidentified 35 has a 30% chance to drop from the quest but if don't get it within 4 attempts, you'll need 10 more Unidentified 34s for every future attempt. Make sure to save 4 Uni 34s for the final quest. If you're very unlucky, you can merge 9 of the ArchFiend Essence Fragments to Swindle's Ripoff Emporium in /tercessuinotlim for the Uni 35.
Use the Reagent Library for details on how to get the following items:
Since you use the Uni 13 in this quest, you'll need to farm a new one each time to accept the quest.