Jeg kan ikke huske, hvilken nøgle der passer til døren.

Because nøgle is a singular common-gender noun in Danish.

• hvilken = for singular common-gender nouns
• hvilket = for singular neuter nouns
• hvilke = for plural nouns

So:

• hvilken nøgle = which key
• hvilket hus = which house
• hvilke nøgler = which keys

Here, der is used because hvilken nøgle is the subject of the embedded clause.

Compare:

• Direct question: Hvilken nøgle passer til døren?
• Embedded question: Jeg kan ikke huske, hvilken nøgle der passer til døren.

In this kind of embedded clause, Danish often inserts der after the question word when that question word is the subject. It is very natural and standard here.

Because hvilken nøgle der passer til døren is a subordinate clause embedded inside the main sentence.

Main clause:

• Jeg kan ikke huske

Subordinate clause:

• hvilken nøgle der passer til døren

Danish normally separates subordinate clauses with a comma, so the comma is required here.

In a main clause, Danish sentence adverbs like ikke usually come after the finite verb.

Here, the finite verb is kan, so:

• Jeg kan ikke huske

This is normal Danish word order.

Compare:

• Jeg husker ikke
• Jeg kan ikke huske

In both cases, ikke comes after the finite verb: husker or kan.

Yes, literally it means cannot remember, but in natural English it is often best translated as don’t remember.

So:

• Jeg kan ikke huske = I can’t remember / I don’t remember

This is a very common Danish way to talk about memory.

Yes. That sentence is also correct.

• Jeg kan ikke huske ... = I can’t remember ...
• Jeg husker ikke ... = I don’t remember ...

The version with kan is very common and often sounds especially natural in everyday speech, but both are good Danish.

Passe til means fit, match, or go with, depending on context.

Here it means that the key is the right one for the door:

• nøglen passer til døren = the key fits the door

The preposition til is part of the expression. Danish often uses verbs together with fixed prepositions, and this is one of them.

Because it means the door, not just a door.

Danish usually marks definiteness by adding an ending to the noun:

• dør = door
• døren = the door

So til døren means for the door or to the door, depending on context. Here it means the specific door being talked about.

The basic form is passe.

• at passe = to fit / to suit / to match
• passer = fits / suits / matches

Danish present tense is often formed by adding -r to the infinitive:

• huske → husker
• passe → passer

Because this is not a direct question. It is an embedded question inside a statement.

Direct question:

• Hvilken nøgle passer til døren?

Embedded question:

• Jeg kan ikke huske, hvilken nøgle der passer til døren.

In embedded questions, Danish does not use the same word order as a main-clause question. That is why the structure looks different from English.

Yes, it functions as a subordinate clause.

Inside that clause:

• hvilken nøgle = the subject being asked about
• der = subject marker in this embedded structure
• passer = verb
• til døren = prepositional phrase

So although it depends on the main clause, it is still a full clause grammatically.

No, not in this sentence.

After an interrogative phrase like hvilken nøgle in this kind of embedded question, Danish uses der, not som:

• Jeg ved ikke, hvem der kommer.
• Jeg kan ikke huske, hvilken nøgle der passer til døren.

Som is used in other relative-clause situations, but not here.

It is a main clause with an embedded indirect question.

Structure:

• Main clause: Jeg kan ikke huske
• Indirect question: hvilken nøgle der passer til døren

So the speaker is not directly asking a question. They are stating that they do not remember the answer.