League-of-legends – How much bonus damage would you deal with a lich bane, a trinity force and an iceborn gauntlet if you used an ability then auto attacked

First, let's clear up a misconception.

Armor Penetration gives you a constant %damage bonus, regardless of how much armor the enemy has see here.

It doesn't matter if your enemy has 300 armor or 15 -- penetrating 10 of that will have the same effect on total DPS increase.

That said, let's take a look at why Armor Penetration is preferred, even with its linearity.

Attack Speed is easy to acquire via items.

Armor Penetration is not.

You can more than match the effects of attack speed runes with a single dagger - 420 gold.

The cheapest item with Armor Penetration is the Brutalizer at 1337 total gold. By this metric alone, Armor penetration runes are worth approximately three times that of attack speed runes.

It is easier to use items to stack Attack Speed over Armor Penetration -- but to get the highest DPS, you want to stack both.

Given that the amount of Armor Penetration available via items is strictly limited, if one's goal is to achieve maximal values for both Armor Penetration and Attack Speed, runes for Armor Penetration are much better than that of Attack Speed.

Like similar ability proc effects (Trinity Force, Sheen, Iceborne Gauntlet), the internal cooldown starts when you attack and consume the damage proc. Therefore, Kayle casting Reckoning in this scenario will not proc Lichbane.