Learn Icelandic: Þeir þekkja ekkert leynifélag nema þetta. - They know no secret society except this one.

When nema acts as a preposition meaning "except", it does not force a specific case of its own. Instead, the word that follows it copies the case of the thing it is being excepted from (the antecedent). Because leynifélag is the direct object of þekkja and is in the accusative case, þetta must also take the accusative to match it.

Why use þekkja instead of vita for "know" in this sentence?

Þekkja is used for being acquainted or familiar with a person, place, or specific entity like a secret society. Vita is used for knowing facts, knowing that something exists, or having information. Because they are familiar with the society itself as an entity rather than just knowing a fact about it, þekkja is the correct verb.

Why do we use ekkert for the word "no" here instead of enginn?

The word for "no" or "none" must agree in gender, number, and case with the noun it modifies. Leynifélag is a neuter noun. Since it is the direct object, it is in the accusative singular. The neuter, accusative, singular form of this negative pronoun is ekkert.

Can I use nema for anything other than "except"?

Yes. While here it works like a preposition meaning "except", nema is also frequently used as a conjunction meaning "unless" (often followed by the subjunctive mood in Icelandic). The context and whether a full clause follows it will usually make the meaning clear.