Learn French: L'exil a beaucoup inspiré le poète. - The exile inspired the poet a lot.

You would use the perfect participle by combining ayant (the present participle of avoir) with the past participle: Ayant beaucoup inspiré le poète... Notice that the adverb beaucoup stays right after the auxiliary verb, just as it does in the passé composé.

Why does beaucoup split the verb (a beaucoup inspiré) instead of going at the end of the sentence?

In French compound tenses like the passé composé, short and common adverbs of quantity or quality (like beaucoup, bien, mal, déjà, and souvent) almost always go between the auxiliary verb (a) and the past participle (inspiré). Saying a inspiré le poète beaucoup would sound unnatural.

Why do we use L' before exil? Couldn't the English translation just be "Exile inspired the poet a lot"?

It absolutely could! However, in French, abstract nouns and general concepts (like exile, love, or life) always require a definite article (le, la, les, or l'). While English frequently drops the article for general ideas, French requires it, which is why we must say L'exil.