Røntgen viste heldigvis, at der ikke var noget alvorligt galt med albuen, men håndleddet skulle have en forbinding.

Here røntgen means the X-ray, the X-ray image, or the X-ray examination. Danish often uses røntgen as a short, practical way of referring to the scan or result when the context is clear.

So Røntgen viste ... means The X-ray showed ...

Also, Røntgen is capitalized here only because it is the first word of the sentence, not because it is a proper name.

Here at means that.

It introduces a subordinate clause after viste:

• Røntgen viste = the X-ray showed
• at der ikke var noget alvorligt galt ... = that there was nothing seriously wrong ...

This is different from the other very common Danish at, which can also mark the infinitive, like English to in at spise = to eat. In this sentence, it is definitely the that-meaning.

Because Danish main clauses normally follow the verb-second pattern.

In Røntgen viste heldigvis ..., the finite verb viste comes early in the sentence, and the adverb heldigvis then follows it.

So the pattern is:

• Røntgen = subject
• viste = finite verb
• heldigvis = adverb

If you moved heldigvis to the front, the verb would still stay in second position:

• Heldigvis viste røntgen ...

So the placement is about normal Danish word order.

This der is a dummy subject, like there in English there was.

So:

• der var = there was
• der ikke var = there was not

The word order changes because this is a subordinate clause introduced by at. In Danish subordinate clauses, words like ikke usually come before the finite verb.

So Danish says:

• at der ikke var ...

not:

• at der var ikke ...

That is a very common and important Danish pattern.

This is built around the very common expression der er noget galt med ..., which means there is something wrong with ...

So:

• noget galt med albuen = something wrong with the elbow
• ikke var noget alvorligt galt med albuen = there was nothing seriously wrong with the elbow

A few details:

• noget here means anything/something
• alvorligt means seriously here, so it functions like an adverb
• galt is the form used in this expression, from gal

Even though gal often means crazy in other contexts, in der er noget galt it means something is wrong.

Danish very often uses the definite form for body parts when it is already obvious whose body part is meant.

So instead of saying something like his elbow or her wrist, Danish often just says:

• albuen = the elbow
• håndleddet = the wrist

This is very natural in medical or physical descriptions.

The endings are different because the nouns have different genders:

• albue is a common-gender noun, so the definite singular is albuen
• håndled is a neuter noun, so the definite singular is håndleddet

So the sentence does not sound incomplete without a possessive. Danish simply does not need one here.

Not in the usual English should have done sense.

Here skulle have means something more like:

• needed a
• was to have
• was supposed to get

So håndleddet skulle have en forbinding means the wrist needed a bandage/dressing, or that a bandage had to be put on it.

In medical Danish, this kind of wording is very common. It often sounds more natural in English as:

• the wrist needed a bandage
• the wrist needed to be bandaged

rather than a literal the wrist should have a bandage.

Because Danish often describes injuries and treatment by making the injured body part the grammatical subject.

So Danish can say:

• Håndleddet skulle have en forbinding

Literally, that is close to the wrist was to have a bandage, but idiomatic English usually prefers:

• the wrist needed a bandage
• the wrist had to be bandaged

This structure is especially common in medical, practical, and report-like language.

En forbinding is a bandage, dressing, or wrap placed on an injury.

The exact English word depends on context:

• bandage if it is a wrapped support
• dressing if it is covering a wound
• sometimes just bandage as a general translation

So in this sentence, en forbinding is the treatment the wrist needed after the X-ray showed there was no serious problem with the elbow.