Når jeg er træt efter arbejdet, har jeg ikke lyst til at lave mad og ordne køkkenet.

Here Når means when in the sense of whenever or every time that.

So the sentence means something like:

Whenever I’m tired after work, I don’t feel like cooking and tidying the kitchen.

A useful contrast:

• når = when / whenever for something general or repeated
• hvis = if for a condition

So:

• Når jeg er træt ... = When/Whenever I’m tired ...
• Hvis jeg er træt ... = If I’m tired ...

In this sentence, når sounds right because it describes a typical situation, not just one possible condition.

This is because Danish main clauses follow the V2 rule: the finite verb must come in the second position.

The first part, Når jeg er træt efter arbejdet, is a subordinate clause placed first in the sentence. After that, the main clause begins, and its finite verb must come first:

• Når jeg er træt efter arbejdet, har jeg ikke lyst ...

If you wrote the main clause by itself, it would be:

• Jeg har ikke lyst til at lave mad ...

But when something else is placed first, Danish changes the order:

• I dag har jeg ikke lyst ...
• Når jeg er træt ..., har jeg ikke lyst ...

So har jeg is normal and required here.

Arbejdet is the definite form of arbejde.

In this sentence, efter arbejdet means after work or more literally after the work / after the working day.

English often says after work without an article, but Danish often uses the definite form in expressions like this. So efter arbejdet is very natural.

You can think of it as referring to the specific workday you have just finished.

At have lyst til at ... is a very common Danish expression meaning:

• to feel like doing something
• to want to do something

So:

• jeg har lyst til at lave mad = I feel like cooking
• jeg har ikke lyst til at lave mad = I don’t feel like cooking

The structure is:

• have lyst til noget = feel like something
• have lyst til at gøre noget = feel like doing something

So in your sentence:

har jeg ikke lyst til at lave mad og ordne køkkenet
= I don’t feel like cooking and tidying the kitchen

Because lyst goes with til.

The full expression is:

• have lyst til noget
• have lyst til at gøre noget

So til belongs to the phrase, and at introduces the infinitive verb.

Compare:

• Jeg har lyst til kaffe. = I feel like coffee.
• Jeg har lyst til at sove. = I feel like sleeping.

So til at is not random — it is the normal pattern after lyst.

Because both verbs belong to the same infinitive structure.

So:

• til at lave mad og ordne køkkenet

means:

• to cook and tidy the kitchen

The at applies to both lave and ordne. Danish often uses at only once before coordinated infinitives.

You could think of it as:

• til at lave mad og (at) ordne køkkenet

But repeating at is usually unnecessary here.

Yes, literally it is make food, but idiomatically it means cook.

This is one of the most common everyday Danish expressions:

• Jeg laver mad. = I’m cooking.
• Kan du lave mad? = Can you cook?

So even though the literal words are make food, the natural English translation is usually cook.

Ordne is a flexible verb that often means:

• tidy
• sort out
• put in order
• take care of

So ordne køkkenet does not only mean one specific action. Depending on context, it may suggest:

• tidying up the kitchen
• cleaning up after cooking
• putting things away
• making the kitchen look orderly again

It is a fairly natural everyday way to say that you deal with the kitchen after using it.

In a main clause, ikke usually comes after the finite verb and before the rest of what is being negated.

Here the finite verb is har, so:

• har jeg ikke lyst ...

This matches normal Danish word order:

• Jeg har ikke lyst.
• Hun kommer ikke i dag.
• Vi spiser ikke kød.

So ikke is in the usual place for a main clause.

Not exactly.

• har ikke lyst til = don’t feel like
• vil ikke = don’t want to / refuse to / am unwilling to

In this sentence, har ikke lyst til is the better choice because it expresses mood or lack of motivation.

Compare:

• Jeg har ikke lyst til at lave mad. = I don’t feel like cooking.
• Jeg vil ikke lave mad. = I don’t want to cook / I refuse to cook.

The second one can sound stronger or more deliberate.

Træt is the correct form here because the subject is singular: jeg.

In this kind of sentence, Danish uses the adjective as a complement after er:

• Jeg er træt. = I am tired.
• Hun er træt. = She is tired.
• Vi er trætte. = We are tired.

So the important contrast is mainly:

• singular: træt
• plural: trætte

That is why jeg er træt is correct.

No. Danish normally requires an explicit subject in the main clause.

So even though the subject is the same in both parts, you still say:

• Når jeg er træt efter arbejdet, har jeg ikke lyst ...

Not:

• Når jeg er træt efter arbejdet, har ikke lyst ...

The first clause and the main clause each need their own grammar, and the main clause must still have a subject: jeg.