We do, though. United Mongol Empire was a relatively short-lived state which quickly broke up into several independent realms, each controlling different parts of the territory: for example, The Golden Horde took firm grip on Eastern European plains, Caucasus and parts of Western Asia, and Yuan Dynasty became in control over China. More over, after these successors broke up into more shards, these shards in turn gave birth to new states, such as Crimean Khanate or Dzungar Khanate. The misconception is here to judge Mongol Empire as a long, coherent civilization lasting for hundreds of years which was never true as different fragments of once united realm quickly fell under influence of their neighbours: for example, Ilkhanate quickly adopted some parts of the unified Iranian culture.

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That's why when we talk about Mongolian legacy, we don't strictly mean only short period of the unified empire but entirety of their conquered territories which produced many works of architecture, poetry and painting. Each of them possessed somewhat unique style: Golden Horde had big Islamic influence while traditional Chinese structure impacted Yan Dynasty. We do have many artifacts left from Mongol successor states in different parts of the world, such as:

1. Khan's mausoleum, 14th century, Bolgar, modern day Russia.
2. Cloud platform, 14th century, Yan Dynasty, modern day China.
3. Stone statue of a turtle, 13th century, Mongol Empire, modern day Mongolia.

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So saying that Mongols didn't leave any legacy is incorrect: they pretty much did, we just aren't used to think about different Mongolian-found states as a unified "nomadic" culture since they quickly changed depending on cultural influence of their neighbours, becoming very distinct from each other.