Treba mi drugi ključ za sobu.

Both are possible, but they’re different constructions:

• Treba mi + NOMINATIVE literally means “(it) is needed to me,” with an impersonal verb and a dative experiencer: Treba mi drugi ključ (od/za sobu).
• Trebam + ACCUSATIVE is the personal form “I need”: Trebam drugi ključ (od/za sobu).

In everyday Croatian, the impersonal Treba mi… is very common and sounds natural and neutral. Trebam… is also accepted and widely used. Choose freely; nuance is minor (Treba mi can sound a touch softer/less direct).

Mi is the unstressed dative singular of the pronoun “I” (ja), meaning “to me/for me.” It marks the experiencer/beneficiary with this verb: Treba mi… = “(something) is needed to me.”

• Dative full (stressed) form for emphasis: meni (e.g., Meni treba… “I, specifically, need…”).
• Don’t confuse with accusative me/mene, used as a direct object with other verbs.

Mi is a clitic (an unstressed short form) and Croatian clitics prefer the “second position” in the clause. The default placement is after the first stressed element:

• Treba mi drugi ključ za sobu.

You can move things around for focus/emphasis, but the clitic still wants second position:

• Za sobu mi treba drugi ključ.
• With emphasis on the person, use the full form first: Meni treba drugi ključ za sobu.

With the impersonal Treba mi, the “needed thing” is in the nominative: drugi ključ. You would not use genitive drugog ključa here.

Compare:

• Impersonal: Treba mi novi prijatelj. (nominative; note it’s not “prijatelja”)
• Personal: Trebam novog prijatelja. (accusative)

In our sentence, even with the personal form, masculine inanimate accusative looks like nominative: Trebam drugi ključ.

• Drugi can mean “second (in order)” or “the other (of two).” In many contexts it also works as “another (different)” key.
• Još jedan literally means “one more,” emphasizing an additional item without implying it’s the other of two.

Nuance:

• If you just want an extra key: Treba mi još jedan ključ (od/za sobu).
• If you mean “a different key” or “the second key (of two)”: Treba mi drugi ključ (od/za sobu).

Za takes the accusative to express purpose/target. Soba (fem.) becomes sobu in the accusative singular:

• za sobu = “for the room (intended for use in that room).”

Both occur, with a nuance:

• ključ od sobe (genitive) = “the key to the room,” i.e., the room’s key (ownership/connection). This is very idiomatic with “key.”
• ključ za sobu (accusative) = “a key intended for the room” (purpose).

In practice, when talking about hotel room keys, ključ od sobe is very common: Treba mi drugi ključ od sobe.

Some polite options:

• Molim vas, mogu li dobiti još jedan/drugi ključ od sobe?
• Možete li mi, molim vas, dati još jedan/drugi ključ od sobe?
• Softened need: Trebao/Trebalo bi mi još jedan/drugi ključ od sobe, molim vas.

Ključ is masculine. Useful forms:

• Nominative sg.: ključ
• Accusative sg. (inanimate): ključ (same as nominative)
• Genitive sg.: ključa
• Nominative pl.: ključevi
• Accusative pl. (inanimate): ključeve
• Genitive pl.: ključeva

Examples:

• Treba mi ključ.
• Trebam dva ključa / Trebaju mi dva ključa.

Both are heard:

• Impersonal pattern (very common): Treba mi dva ključa.
• Personal agreement with the plural subject: Trebaju mi dva ključa.

In everyday speech, both are acceptable. The impersonal treba is extremely frequent.

Yes:

• Treba mi = “I need (something).” Focus on necessity.
• Nedostaje mi = “I’m missing/lacking (something)” or “I miss (someone).” Focus on absence.

For a hotel key, both can work, but Treba mi (još jedan/drugi) ključ od sobe is the straightforward “I need” request.

Yes; you can shift elements for emphasis while keeping the clitic second:

• Neutral: Treba mi drugi ključ od/za sobu.
• Emphasis on “for the room”: Za sobu mi treba drugi ključ.
• Emphasis on “me”: Meni treba drugi ključ od/za sobu.

All are fine; choose based on what you want to highlight.