Der mangler en knap på min skjorte, så jeg finder en nål og tråd.

Here der is a dummy subject, similar to the there in English sentences like There is a problem.

So Der mangler en knap literally looks like There lacks a button, but the natural meaning is A button is missing.

Important point:

• der here does not mean there as a place.
• It is just part of a very common Danish pattern for saying that something is missing, available, happening, etc.

Mangle usually means to be missing, to lack, or to need.

In this sentence:

• Der mangler en knap = A button is missing

This is different from savne, which is the verb you use for missing someone emotionally:

• Jeg savner dig = I miss you
• Der mangler en knap = A button is missing

So for an object that is absent, mangle is the right verb.

Because the sentence is introducing the button as an unspecified missing button, not as a previously identified one.

• en knap = a button
• knappen = the button

So Der mangler en knap på min skjorte means There is a button missing on my shirt / A button is missing from my shirt.

If you had already been talking about a specific button, then knappen could make sense in another context.

Danish often uses på with things that are attached to a surface or part of clothing.

So:

• en knap på min skjorte = literally a button on my shirt

Even though English often prefers missing from my shirt, Danish normally uses på here because the button is something located on the shirt.

This is one of those places where Danish follows its own preposition pattern rather than matching English exactly.

In Danish, a possessive like min, din, hans, vores normally replaces the article.

So you say:

• min skjorte = my shirt
• not min skjorten
• not den min skjorte

This is very similar to English:

• my shirt
• not the my shirt

So the structure is completely normal Danish.

Because så here works as a coordinating conjunction meaning so.

When two main clauses are joined by a coordinating conjunction, the second clause keeps normal main-clause word order:

• Der mangler en knap ..., så jeg finder ...

That means:

• subject + verb → jeg finder

Compare that with an adverb like derfor:

• Der mangler en knap. Derfor finder jeg en nål og tråd.

Here derfor takes first position in the clause, so Danish does inversion:

• Derfor finder jeg ...

So in your sentence, så is the reason the order is jeg finder, not finder jeg.

Yes, but in this kind of context finder can mean something like:

• find
• get hold of
• go and get
• take out

So jeg finder en nål og tråd is not necessarily about a long search. It often just means the speaker goes to get those things because they need them.

In English, you might translate it more naturally as:

• so I get a needle and thread rather than
• so I find a needle and thread

But the Danish verb finder is still perfectly normal here.

Because tråd here is being used as an uncountable material noun, meaning thread in general.

So:

• en nål = a needle
• tråd = thread

This is very natural in Danish, especially in the common pairing nål og tråd.

If you said en tråd, that would more likely suggest a single thread / a strand of thread, which is usually not the main idea here.

So en nål og tråd is the natural way to say a needle and thread.

Yes, it is a very common combination, much like English needle and thread.

Because the two items naturally belong together for sewing, Danish often treats them almost as a standard pair. That is another reason the sentence sounds very natural as:

• jeg finder en nål og tråd

rather than repeating the article before both nouns.

Yes, that is understandable, but Der mangler en knap på min skjorte is usually the more natural and idiomatic way to say it.

Compare the focus:

• Der mangler en knap på min skjorte
  Neutral, very natural way to state that a button is missing.

• Min skjorte mangler en knap
  Also possible, but it can sound a bit more literal or less idiomatic.

So as a learner, Der mangler en knap på min skjorte is a very good pattern to remember.

Here they are:

• en knap → knappen
• en skjorte → skjorten
• en nål → nålen
• en tråd → tråden

So all of these are common gender nouns, which is why their basic singular form takes en.

In the sentence, though, tråd appears without en because it means thread as a material, not one thread as a single countable object.