Vi diskuterede problemet længe, men til sidst fandt vi ud af, at det hele skyldtes en gammel misforståelse.

Because problem is the indefinite form, meaning a problem or just problem as a dictionary form.

The noun is neuter: et problem.
The definite singular form is problemet, meaning the problem.

So:

• et problem = a problem
• problemet = the problem

They are all in the past tense.

• diskuterede = discussed
• fandt = found
• skyldtes = was due to / was caused by

So the whole sentence tells a completed event in the past.

Yes. The infinitive is at diskutere = to discuss. Its past tense is diskuterede, which is a regular weak verb pattern in Danish.

So:

• at diskutere = to discuss
• diskuterer = discuss / discusses
• diskuterede = discussed

Længe means for a long time or for long.

In Danish, you can often use længe directly as an adverb without adding a separate word for for. So Vi diskuterede problemet længe means We discussed the problem for a long time.

A close alternative would be i lang tid, which also means for a long time.

Til sidst means in the end, finally, or eventually.

It shows that something happened after some time or after a process. In this sentence, it tells us that after discussing the problem for a long time, they finally reached a conclusion.

Fandt ud af is the past tense of finde ud af, which is a very common Danish expression meaning:

• find out
• figure out
• sometimes realize

In this sentence, fandt vi ud af means something like we found out or we figured out.

Because finde ud af is a multi-word verb, and in a main clause the finite verb often separates from the rest of the expression.

So the basic verb is:

• at finde ud af = to find out / figure out

But in the past tense main clause, it becomes:

• vi fandt ud af = we found out

The part ud af stays later in the clause, while the finite verb fandt moves into the normal main-clause verb position.

This is because Danish main clauses normally follow the V2 rule, meaning the finite verb comes in the second position.

Here, til sidst takes the first position. That means the verb must come next:

• Til sidst fandt vi ud af ...

So even though vi is the subject, it comes after the verb because something else has been placed first.

Compare:

• Vi fandt ud af det.
• Til sidst fandt vi ud af det.

Here at means that and introduces a subordinate clause.

So:

• vi fandt ud af, at ... = we found out that ...

In English, that is often omitted, but in Danish at is very commonly kept.

Det hele means the whole thing, all of it, or sometimes everything depending on context.

In this sentence, it refers back to the whole problem or situation they were discussing. So at det hele skyldtes ... means that the whole thing was due to ...

Because the verb here is skyldes, not just skylde.

Skyldes means be due to or be caused by. Its past tense is skyldtes.

So:

• Det skyldes en fejl. = It is due to a mistake.
• Det skyldtes en fejl. = It was due to a mistake.

The -s is part of the verb form here, not a plural ending.

Because misforståelse is a common-gender noun, so it takes en in the indefinite singular:

• en misforståelse = a misunderstanding

The adjective matches that form:

• en gammel misforståelse = an old misunderstanding

Compare:

• en gammel bog = an old book
• et gammelt problem = an old problem

So gammel is the correct adjective form for an indefinite singular common-gender noun.

That is a punctuation choice found in one accepted Danish comma system.

The sentence has:

• ..., men til sidst fandt vi ud af, at ...

The comma before men is normal because it separates main clauses. The comma before at depends on the comma style being used; many writers include it, while others would write fandt vi ud af at ... without that comma.

So the sentence is completely normal, but you may also see a version without the comma before at.