Nøglekortet virker også ved døren i trappeopgangen.

Because nøglekortet is the definite form: the key card.

In Danish, the definite article is usually added to the end of the noun:

• et nøglekort = a key card
• nøglekortet = the key card

The ending -et shows that nøglekort is a neuter noun.

Yes. Danish, like English, often makes compound nouns, but Danish usually writes them as one word.

So:

• nøgle = key
• kort = card
• nøglekort = key card

This is very normal in Danish. English sometimes writes similar things as two words, but Danish often prefers one compound word.

Here virker means works or functions.

It is the present tense of at virke.

• Det virker = It works
• Nøglekortet virker = The key card works

Be aware that at virke can also mean seem/appear in other contexts, but in this sentence it clearly means function properly.

This is normal Danish word order in a main clause.

In a standard statement, the finite verb usually comes in the second position. After that, adverbs like også often come before other sentence elements.

So:

• Nøglekortet = subject
• virker = finite verb
• også = adverb

That gives:

• Nøglekortet virker også ...

This is very natural Danish.

Ved usually means by, at, or near.

So ved døren means something like:

• at the door
• by the door
• near the door

In this sentence, it suggests the key card works at that door/access point. In practical English, you might translate it simply as at the door.

Because the two prepositions express different kinds of location.

• ved døren = at/by the door
  This means near a specific point or object.
• i trappeopgangen = in the stairwell / in the staircase area
  This means inside that area or space.

So Danish is distinguishing between:

1. the exact point where the card works: ved døren
2. the larger area where that door is located: i trappeopgangen

Because both refer to specific things:

• døren = the door
• trappeopgangen = the stairwell / the entrance stairway

Danish often uses the definite form when the speaker assumes the listener knows which door or which stairwell is meant, for example in a hotel, apartment building, or office building.

It is a compound noun:

• trappe = stairs / staircase
• opgang = entrance, stairwell, stairway section of a building

Together, trappeopgang usually refers to the shared staircase/stairwell area in a building.

Then the definite ending is added:

• en trappeopgang = a stairwell
• trappeopgangen = the stairwell

Depending on context, English might translate it as the stairwell, the staircase, or the entrance stairway.

Most naturally, it describes døren:
the door in the stairwell.

So the structure is understood as:

• ved døren i trappeopgangen = at the door in the stairwell

Grammatically, it is attached to the noun phrase with døren, not mainly to virker. In other words, it tells you which door.

Not as a normal main clause.

In Danish main clauses, the finite verb usually has to be in second position, so the correct order is:

• Nøglekortet virker også ved døren i trappeopgangen.

But your version can work inside a subordinate clause, for example:

• Jeg ved, at nøglekortet også virker ved døren i trappeopgangen.
  = I know that the key card also works at the door in the stairwell.

So:

• main clause: verb in second position
• subordinate clause: the adverb også can come before the verb phrase differently

There are two genders represented here:

• et nøglekort → neuter
  definite: nøglekortet
• en dør → common gender
  definite: døren
• en trappeopgang → common gender
  definite: trappeopgangen

So the definite endings match the gender:

• -et for many neuter nouns
• -en for many common-gender nouns

This is why the sentence has both -et and -en endings.