Hun siger, at hun heller ikke orker at gå ud, fordi hun har haft en lang arbejdsdag.

Here at introduces a subordinate clause, like English that in She says that...

So:

• Hun siger = She says
• at hun heller ikke orker... = that she also doesn’t feel up to...

In English, that is often optional, but in Danish at is very common in this structure.

Because this is a subordinate clause after at.

In Danish:

• Main clauses usually follow verb-second word order.
• Subordinate clauses do not.

That means sentence adverbs like ikke, aldrig, heller ikke usually come before the finite verb in a subordinate clause.

So:

• Hun orker heller ikke at gå ud. = main clause
• ... at hun heller ikke orker at gå ud. = subordinate clause

This is one of the most important word-order patterns in Danish.

Heller ikke means not either or neither.

It is used in negative contexts to add something that is also negative.

Examples:

• Jeg kommer ikke. = I’m not coming.
• Hun kommer heller ikke. = She isn’t coming either.

So in your sentence, hun heller ikke orker suggests that someone else probably also doesn’t feel like going out, and she is adding herself to that negative idea.

Because Danish normally uses:

• også for positive addition: also / too
• heller ikke for negative addition: not either / neither

So:

• Jeg vil også med. = I also want to come.
• Jeg vil heller ikke med. = I don’t want to come either.

Også ikke is generally not the natural choice here.

Not exactly.

At orke means something like:

• to have the energy to
• to feel up to
• to be able to face doing something

So hun orker ikke at gå ud is not simply she doesn’t want to go out. It is closer to:

• she can’t be bothered going out
• she doesn’t have the energy to go out
• she’s not up for going out

It often suggests tiredness, low energy, or mental exhaustion.

That at is the infinitive marker, equivalent to English to.

So:

• at gå ud = to go out

After orke, Danish normally uses an infinitive:

• orke at gøre noget = have the energy to do something

So:

• hun orker at gå ud = she feels up to going out / she has the energy to go out

Literally, gå ud means go out.

In this sentence, it most likely means:

• going out socially
• going out in the evening
• leaving home to do something outside

It does not necessarily mean walking. Danish often uses gå ud in the same broad way English uses go out.

Har haft is the present perfect:

• har = present tense of have
• haft = past participle of have

So hun har haft en lang arbejdsdag means she has had a long workday.

This tense is used because the workday happened earlier, but its effect is still relevant now: she is tired now because of it.

Yes, that is possible in many contexts.

The difference is roughly:

• hun har haft en lang arbejdsdag = she has had a long workday
  Focus on present relevance; the long day explains how she feels now.
• hun havde en lang arbejdsdag = she had a long workday
  More like a simple past fact.

In this sentence, har haft sounds very natural because it connects the earlier workday to her present lack of energy.

Those commas mark subordinate clauses.

The sentence contains:

• at hun heller ikke orker at gå ud
• fordi hun har haft en lang arbejdsdag

In Danish, commas are often used to separate subordinate clauses from the main clause. So the commas help show the structure of the sentence.

They are grammatical punctuation marks, not pauses you must strongly pronounce.

Fordi means because and introduces the reason.

So:

• Hun siger, at hun heller ikke orker at gå ud = She says that she also doesn’t feel up to going out
• fordi hun har haft en lang arbejdsdag = because she has had a long workday

It connects her lack of energy with its cause.

Because each clause needs its own subject.

The sentence has:

• Hun siger = main clause
• at hun heller ikke orker at gå ud = subordinate clause

So the second hun is the subject of the subordinate clause.

English works the same way:

• She says that she...

Even if both refer to the same person, Danish still repeats the pronoun.

Because arbejdsdag is a common gender noun, so it takes en.

In the indefinite singular:

• common gender noun: en lang arbejdsdag
• neuter noun: et langt ...

So here:

• en because arbejdsdag is an en-word
• lang because it is indefinite singular common gender

If it were definite, you would say:

• den lange arbejdsdag = the long workday

It is a compound noun, which is very common in Danish.

It is made from:

• arbejde = work
• linking -s-
• dag = day

So arbejdsdag literally means work-day, i.e. workday.

Danish forms many nouns this way, so getting used to compounds is very useful for reading and vocabulary building.