Running demon blades does not remove your melee attacks, you still do them as normal so you would still get the 10% bonus. The issue is your melee are likely sitting below 4% of your damage, so the proc is worth (using a lot of mine) approximately 0.3% dps - not worth any significant ilvl downgrade for us.
Nightblooming frond's increasing melee damage is it's own damage source - recursive strikes - and therefore does not get affected by the ring.