Hun leder efter sine handsker, men de ligger i frakkens lomme.

Because sine is the reflexive possessive form. Danish uses sin / sit / sine when the thing belongs to the subject of the clause.

Here, the subject is Hun (she), so:

• Hun leder efter sine handsker = She is looking for her own gloves

If you said hendes handsker, it would usually mean:

• her gloves = another woman’s gloves, not necessarily the subject’s

A very important detail: sin / sit / sine agrees with the thing owned, not with the owner:

• sin jakke = her/his own jacket
• sit hus = her/his own house
• sine handsker = her/his own gloves

So sine is used here because handsker is plural.

At lede efter is a fixed expression meaning to look for or to search for.

So:

• Hun leder efter sine handsker = She is looking for her gloves

Grammatically:

• leder = present tense of at lede
• efter = for/after, but in this expression you should learn the whole phrase together

It works a bit like an English phrasal verb. You should think of lede efter as one unit of meaning.

Because de refers back to handsker, which is plural.

So:

• de = they

Here it is the subject of the second clause:

• men de ligger ... = but they are lying / but they are ...

Danish distinguishes between:

• de = subject form (they)
• dem = object form (them)

So this sentence needs de, not dem, because the gloves are doing the action/state of ligger.

Danish often uses a position verb where English would simply use be.

Here, ligger literally means lie / are lying. It is very natural for things like gloves, keys, papers, clothes, etc. when they are resting somewhere.

So:

• de ligger i frakkens lomme = literally they are lying in the coat’s pocket

In English, we would usually just say they are in the coat pocket, but Danish often prefers ligger in this kind of situation.

Very roughly:

• ligge = lie, be lying
• stå = stand, be standing
• sidde = sit, be situated/fitted in some contexts

Frakkens is the genitive form of frakken.

Step by step:

• frakke = coat
• frakken = the coat
• frakkens = the coat’s

So:

• frakkens lomme = the coat’s pocket / the pocket of the coat

Danish usually forms possession by adding -s to the noun, much like English ’s, but without an apostrophe in normal cases.

Examples:

• mandens bil = the man’s car
• pigens bog = the girl’s book
• frakkens lomme = the coat’s pocket

Because after a genitive owner like frakkens, the thing possessed is usually in the indefinite form in Danish.

So Danish says:

• frakkens lomme

not usually:

• frakkens lommen

This is different from English, where we say the coat’s pocket with the.

A useful pattern is:

• mandens hus = the man’s house
• barnets cykel = the child’s bike
• frakkens lomme = the coat’s pocket

So even though it is definite in meaning, Danish normally uses the possessed noun without the definite ending here.

Because men (but) is a coordinating conjunction, and after it Danish normally keeps normal main-clause word order:

• subject + verb

So:

• men de ligger ...

is correct.

Danish inversion happens after certain sentence elements are placed first, for example:

• I frakkens lomme ligger de = In the coat pocket they are

But after men, you normally just continue with ordinary main-clause order:

• Hun leder ..., men de ligger ...

Handsker is the plural indefinite form of handske (glove).

The forms are:

• en handske = a glove
• handsken = the glove
• handsker = gloves
• handskerne = the gloves

So in this sentence:

• sine handsker = her own gloves

Because the sentence has two main clauses:

• Hun leder efter sine handsker
• de ligger i frakkens lomme

They are joined by men (but), and Danish normally places a comma between such clauses.

So the comma helps show the structure:

• first statement: she is looking for them
• contrasting statement: but they are actually in the coat pocket

Yes. This sentence is perfectly good Danish, but there are other natural ways to express similar ideas.

For example:

• Hun leder efter sine handsker, men de ligger i lommen på frakken.
• Hun leder efter sine handsker, men de ligger i frakkelommen.

These all mean roughly the same thing.

The differences are mostly stylistic:

• i frakkens lomme = in the coat’s pocket
• i lommen på frakken = in the pocket of the coat
• i frakkelommen = in the coat pocket

All are understandable and natural in the right context.

Leder efter is very natural here. It is the everyday way to say looking for something that is missing.

You might also see:

• søger efter

But søger efter can sound a bit more formal or more like searching than ordinary everyday looking for.

So for lost gloves, leder efter is the most natural choice.