Learn Polish: Pomimo wojny Polska odzyskała niepodległość w dwudziestym wieku. - Despite the war, Poland regained independence in the twentieth century.

To say 'in a specific century', Polish uses the preposition w followed by the Locative case. The ordinal number dwudziesty (twentieth) changes to its Locative form dwudziestym, and the noun wiek (century) becomes wieku.

What case does the word wojny take after pomimo?

It takes the Genitive case. The preposition pomimo (despite) always requires the Genitive, which is why the base feminine noun wojna changes to wojny.

Why is the verb odzyskała used instead of the imperfective odzyskiwała?

Regaining independence is being described as a completed, successful action, so we need the perfective verb odzyskać. The ending is feminine to match the subject of the sentence, Polska, which is a feminine noun.

Why doesn't the word niepodległość change its ending if it is the object of the sentence?

It is actually in the Accusative case, but feminine nouns that end in a consonant, like niepodległość (independence), do not change their form in the Accusative. They look exactly the same as they do in their base Nominative dictionary form.