En lille detalje i kontrakten ændrede hendes beslutning om lejligheden.

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

• en detalje = a detail

If it were a neuter noun, it would take et instead.

For example:

• en detalje
• et problem

So en lille detalje is correct because detalje is an en-word.

Yes, adjectives often change form in Danish, but lille is a bit special.

In this sentence:

• en lille detalje

lille is the correct form before a singular noun here. It is one of the common adjective forms learners meet early on.

Compare:

• en stor detalje = a big detail
• et stort problem = a big problem

With lille, Danish often uses:

• en lille detalje
• et lille problem
• små detaljer = small details

So lille is just the normal form here.

Because Danish usually puts the definite article at the end of the noun.

So:

• en kontrakt = a contract
• kontrakten = the contract

The same pattern appears with:

• en lejlighed = an apartment
• lejligheden = the apartment

This is one of the biggest differences from English. Instead of a separate the, Danish often adds -en, -et, or another definite ending to the noun.

Because the two prepositions mean different things.

• i kontrakten = in the contract
• om lejligheden = about / concerning the apartment

So the sentence means that a detail in the contract changed her decision about the apartment.

This is natural Danish:

• i is used for something inside a document, place, text, etc.
• om is used for something concerning a topic or subject

Because sin/sit/sine is only used when the possessor is the subject of the clause.

Here, the subject is:

• En lille detalje i kontrakten

That means the subject is not she. So Danish cannot use sin to refer to her. It must use:

• hendes beslutning = her decision

Compare:

• Hun ændrede sin beslutning. = She changed her decision.
  Here hun is the subject, so sin is correct.

But in your sentence:

• En lille detalje ... ændrede hendes beslutning ...
  The subject is en lille detalje, so hendes is required.

Because possessives like min, din, hans, hendes, vores usually replace the article.

So you say:

• hendes beslutning = her decision

not:

• den hendes beslutning
• en hendes beslutning

This works much like English, where you say her decision, not the her decision.

Ændrede is the past tense of the verb ændre.

The basic forms are:

• at ændre = to change
• ændrer = changes / is changing
• ændrede = changed
• ændret = changed (past participle)

So in this sentence, ændrede tells you the action happened in the past:

• En lille detalje ... ændrede ... = A small detail ... changed ...

The structure is:

• En lille detalje i kontrakten = subject
• ændrede = finite verb
• hendes beslutning om lejligheden = object

So the sentence follows a very normal Danish main-clause pattern:

subject + verb + object

Danish is also a V2 language, which means the finite verb normally comes in the second position in a main clause. In this sentence, the subject comes first, so the verb comes right after it.

It means the decision regarding the apartment or the decision about the apartment.

It does not necessarily tell you what the decision was. It just says that she had some decision connected to the apartment—for example, whether to rent it, buy it, accept it, or reject it.

So om lejligheden gives the topic of the decision.

It is understandable and basically natural. A Dane could certainly say it.

That said, another very common way to express the same idea would be something like:

• En lille detalje i kontrakten fik hende til at ændre sin beslutning om lejligheden.

Literally, that means:

• A small detail in the contract made her change her decision about the apartment.

This version makes the cause-and-effect relationship a little more explicit. But your original sentence is still grammatically fine and clear.