Did some testing. Used it on the lothric knight at dragon barracks bonfire. Did 23 damage per tick regardless of items and upgrades. 1380 damage total. Duration also unaffected by items. But the buildup got weaker when I switched to an unupgraded flame. So that seems to be the only thing you can boost.
Tested these rings: Great Swamp Ring, Witch's Ring, Lingering Dragoncrest Ring, all Clutch Rings, Lloyd's Sword Ring, Red Tearstone Ring
Another note: the spell goes through walls and does not cause the enemy to notice you. Perfect for cheese