Learn Polish: Po nieprzespanej nocy funkcjonuję gorzej cały kolejny dzień. - After a sleepless night, I function worse the whole next day.

This is the "bare accusative of duration." In Polish, when you state exactly how long an action lasts (in this case, "the whole next day"), you put the time phrase directly into the accusative case without any preposition. It acts just like "for the whole next day" in English.

How is the word nieprzespanej formed?

It is built from the perfective verb przespać (to sleep through). It is turned into the passive participle przespany (slept through) and then given the negative prefix nie- to mean "sleepless" (literally: "not-slept-through"). It takes the Locative case here (nieprzespanej nocy) because it is triggered by the preposition po.

Why use the adverb gorzej instead of an adjective like gorszy?

English speakers often use adjectives to describe their state (e.g., "I function worse" or "I feel bad"). In Polish, however, you are describing the action of the verb funkcjonuję (how do you function?), so you must use an adverb. Gorzej is the comparative adverb form of źle (badly).

Could I say cały następny dzień instead of cały kolejny dzień?