Hun har en gammel trøje med korte ærmer på i køkkenet.

In everyday Danish, at have ... på is the most common way to say that someone is wearing something.

So:

• Hun har en gammel trøje på = She is wearing an old sweater

Literally, it is close to English she has an old sweater on. Danish does have other ways to express this idea, but have ... på is the normal, natural everyday choice.

På is part of the expression have på, which means to have on / to be wearing.

In Danish, the thing being worn usually goes between har and på:

• Hun har en trøje på
• Han har sko på

So the sentence structure is very normal. English can do something similar:

• She has a sweater on

The extra description med korte ærmer stays with trøje, so på ends up even farther to the right.

Because trøje is a common-gender noun in Danish, so it takes en, not et.

• en trøje = a sweater
• et hus = a house

The adjective must match that grammar pattern:

• common gender singular: en gammel trøje
• neuter singular: et gammelt hus

So gammel is the correct form here because trøje is an en-word.

In the indefinite singular with a common-gender noun, adjectives usually use their basic form.

So:

• en gammel trøje = an old sweater

You get gamle in other situations, for example:

• plural: gamle trøjer = old sweaters
• definite forms: den gamle trøje = the old sweater

So here, gammel is correct because the noun phrase is:

• singular
• indefinite
• common gender

Because ærmer is plural.

• et ærme = a sleeve
• ærmer = sleeves

In Danish, adjectives before plural nouns usually take -e, so:

• korte ærmer = short sleeves

Compare:

• et kort ærme = a short sleeve
• korte ærmer = short sleeves

So med korte ærmer literally means with short sleeves.

It describes the sweater.

So the structure is:

• en gammel trøje med korte ærmer = an old sweater with short sleeves

The phrase med korte ærmer comes right after trøje, which makes it clear that it belongs to the noun trøje.

It does not mean that the woman somehow has short sleeves by herself; it is a description of the clothing item.

Because Danish usually marks definiteness by adding it to the end of the noun.

• et køkken = a kitchen
• køkkenet = the kitchen

So:

• i køkkenet = in the kitchen

This is one of the big differences from English. Instead of a separate word like the, Danish often uses a suffix:

• bogen = the book
• huset = the house
• køkkenet = the kitchen

Normally, it means she is in the kitchen while wearing the sweater.

In other words, the most natural reading is:

• She is in the kitchen, wearing an old sweater with short sleeves.

Grammatically, Danish can sometimes leave this kind of attachment slightly open, just like English can. But in normal interpretation, the location phrase refers to the person and the situation, not to the sweater by itself.

Yes. Danish often moves time or place phrases to the front of the sentence.

For example:

• I køkkenet har hun en gammel trøje med korte ærmer på.

That is still correct Danish. The important thing is that Danish main clauses usually follow the V2 rule: the finite verb comes second.

So if i køkkenet moves to the front, har stays in second position, and hun comes after it.

No, but it is the most everyday, natural one.

Other possibilities include:

• Hun er iført en gammel trøje med korte ærmer.
  More formal, more written-style. Roughly She is dressed in / is wearing...

• Hun går med en gammel trøje med korte ærmer.
  This can suggest something more habitual, like she goes around wearing or she wears that kind of thing.

So for ordinary spoken Danish, Hun har ... på is the best pattern to learn first.