Vi letade efter hennes glasögon, men de fanns ingenstans.

Because leta efter is the normal Swedish expression for look for.

• leta = to search, look around
• leta efter något = to look for something specific

So:

• Vi letade efter hennes glasögon = We were looking for her glasses

If you leave out efter, the sentence can sound incomplete or change meaning.

Because sin/sitt/sina refers back to the subject of the clause.

Here, the subject is vi (we), not hon (she). So the glasses belong to some other female person, and Swedish uses hennes.

• Vi letade efter hennes glasögon = We were looking for her glasses.
• Hon letade efter sina glasögon = She was looking for her own glasses.

So hennes is correct because the owner is not the subject vi.

After a possessive word like min, din, hans, hennes, vår, deras, Swedish normally uses the noun without the definite ending.

So:

• hennes glasögon = her glasses
• not hennes glasögonen

This is different from English, where you just say her glasses and do not visibly change the noun. In Swedish, the possessive already makes the noun definite enough, so the extra definite ending is usually not added.

In normal Swedish, glasögon behaves like a plural noun, much like English glasses.

That is why the sentence later uses de:

• glasögon → de
• her glasses → they

If you want to talk about one pair, Swedish still usually says:

• ett par glasögon = a pair of glasses

There is no common everyday singular equivalent of a glass-eye-wear item.

Because de refers back to glasögon, which is grammatically plural.

So:

• glasögon = glasses
• de = they

The second clause means:

• men de fanns ingenstans = but they were nowhere to be found

Swedish often avoids repeating the noun if a pronoun is clear.

Because de is the subject of the clause:

• de fanns ingenstans

Here, de is doing the action of fanns in a grammatical sense, so Swedish uses the subject form de.

A simple rule:

• de = they
• dem = them

In everyday speech, many Swedes pronounce both de and dem as dom, but in standard writing this sentence needs de.

Fanns is the past tense of finnas, which often means to exist, to be, or to be found.

In this sentence:

• de fanns ingenstans = they were nowhere / they were nowhere to be found

This sounds more natural in Swedish than using vara here.

Compare:

• Det finns en bok på bordet = There is a book on the table
• Boken fanns inte där = The book wasn’t there / couldn’t be found there

So fanns gives the idea of existence or presence in a place, not just a basic were.

Ingenstans is an adverb meaning nowhere.

It is commonly written as one word:

• någonstans = somewhere / anywhere
• ingenstans = nowhere

So:

• de fanns ingenstans = they were nowhere to be found

It is a very common word in Swedish and is useful to learn as a fixed item.

Because men (but) is a coordinating conjunction, so the clause after it keeps normal main-clause word order.

Normal Swedish main-clause order is usually:

• subject + verb

So:

• de fanns ingenstans

This is different from some other conjunctions that introduce subordinate clauses, where the word order changes.

Compare:

• ..., men de fanns ingenstans. = main clause after men
• ..., eftersom de inte fanns där. = subordinate clause after eftersom

So the word order here is completely normal.

In everyday speech, many Swedes pronounce de as dom.

So this sentence is often heard roughly as:

• Vi letade efter hennes glasögon, men dom fanns ingenstans.

But in standard writing, you should still write de, not dom, unless you are using an informal style where dom is accepted.

Yes, you often can, but letade efter is the most natural everyday choice here.

• leta efter = look for
• söka efter = search for

Söka efter can sound a little more formal, deliberate, or systematic, depending on context. For missing glasses in ordinary conversation, letade efter is usually the best fit.