Why Diana doesn't have Luden's as a recommended item?

because protobelt, GLP, lich bane, and nashor's are literally all better as "extra damage" items

It's because Lychbane is better for her, specially since she's AA based. Luden's is better for burst mages than assassins, and the main utility is helping with waveclear.

She can proc Lich Bane more frequently for higher sustained damage (and burst damage past a certain point).

It's true that is a nice item because of the AP stats and bonus movement speed. But Diana's build really comes down with a or . Rod of ages is there to survive against burst champions and having extra mana because she uses a lot of her abilities. And Nasher's tooth synergizes with Diana's passive and allows her to proc it faster because of the bonus Atk spd. Having both of these items is nice but many players chooses one of the two or go with . If you still prefer to build Ludens, save as a last item because you need your and for an heavy ad comp or for AP. Or if you're super ahead like 3-0, you can make it a first item.