Luden's Echo applies on spell hit, not magic damage.
That seems to confuse everyone. Basically put, an on spell-hit effect (Rylai's, Liandry's, Luden's, etc.) is the spell-equivalent of an on attack-hit effect (Frozen Mallet, BotRK, Statikk Shiv, etc.).
If the damage source is a spell--if it comes from an activated Q/W/E/R, is not an attack modifier, and does not say "applies on-hit effects" (referring to attack effects)--then it will proc regardless of the damage type. This is why, for a very long time, people asked Riot for an AD Spell Vamp item (particularly for Riven...*shudder*)
There are a few attack modifiers that can apply spell effects. The ones I can name off-hand are Jax's Empower (W, I assume this is because it can also apply to his Q, making it also a spell effect) and Grandmaster's Might (R passive, your guess is as good as mine as to why), Kayle's Righteous Fury (E, due to coding, it's a spell effect that requires a basic attack to acquire its target), Cho'gath's Vorpal Spikes (E passive, similar reason to Kayle except it's a short-range line skill shot), and Bard's Meeps (passive, I would guess it's because of the charge-based cooldown, but it's a totally blind guess).
Yes, that's right, Bard's Meeps can proc Luden's Echo. And his basic attacks--like any--can proc Statikk Shiv. So he can actually have a pretty impressive burst build on top of a pair of MS items.