Learn Icelandic: Ég klára hvert verk hvern dag.

They are both in the accusative case, but they must agree with the gender of the noun they modify. Verk is a neuter noun, so it takes the neuter accusative form hvert. Dagur (day) is a masculine noun, so it takes the masculine accusative form hvern.

Why are both hvert verk and hvern dag in the accusative case in this sentence?

Hvert verk is the direct object of the verb klára (to finish), which triggers the accusative case. Hvern dag is in the accusative for a different reason: in Icelandic, phrases that describe time frequency, like "every day" or "each year", automatically use the accusative case.

Doesn't hver usually mean "who"? How can I tell when it means "each" or "every"?

It does mean "who" when used as a question word! However, it also acts as an indefinite pronoun meaning "each" or "every". You can tell the difference from the context: when it means "each" or "every", it is almost always attached directly to a noun, just like hvert verk and hvern dag here.

If verk is in the accusative case, why does it look exactly like its dictionary form?