Hun viser ansvar og hensyn, når hun bliver hjemme med sin syge datter.

Viser ansvar og hensyn is a very natural Danish expression meaning shows responsibility and consideration. It focuses on what her action demonstrates.

You could also say Hun er ansvarlig, but that means She is responsible as a more general character description.
Hun viser ansvar og hensyn is more about this specific situation: by staying home, she is behaving in a responsible and considerate way.

Hensyn means consideration, regard, or thought for other people or circumstances.

In this sentence, ansvar og hensyn works as a pair:

• ansvar = responsibility
• hensyn = consideration for others / thoughtfulness

A very common expression is tage hensyn til = to take something/someone into consideration.

Because ansvar and hensyn are abstract nouns here, used in a general sense.

So Danish says:

• vise ansvar
• vise hensyn

rather than something like vise et ansvar or vise et hensyn.

This is similar to English, where we often say show responsibility or show consideration without an article.

Når means when in the sense of:

• something that happens generally
• something that happens whenever a certain situation occurs

Here it describes the circumstance under which she shows responsibility and consideration: when she stays home with her sick daughter.

Compare:

• når = when / whenever
• hvis = if
• da = when, for a specific event in the past

So når fits well because this is presented as a general situation, not one single completed past event.

That is a very common learner question. Blive can mean both:

• become
• stay / remain

In bliver hjemme, it means stays home or remains at home.

So:

• hun bliver træt = she becomes tired
• hun bliver hjemme = she stays home

You have to tell from the context which meaning is intended.

Because Danish often distinguishes between:

• hjem = motion toward home
• hjemme = location at home

So:

• Hun går hjem = She goes home
• Hun er hjemme = She is at home
• Hun bliver hjemme = She stays at home

In this sentence, there is no movement toward home. She is remaining there, so hjemme is correct.

Because sin is the reflexive possessive. It refers back to the subject of the clause.

Here the subject is hun, and the daughter belongs to that same hun, so Danish uses sin:

• Hun bliver hjemme med sin syge datter = She stays home with her own sick daughter

If you said hendes, it would normally suggest someone else’s daughter, not the subject’s own daughter.

This is an important Danish pattern:

• Han elsker sin kone = he loves his own wife
• Han elsker hans kone = he loves his wife, but not his own; someone else’s or a different man’s wife is implied

Because the adjective comes after a possessive determiner: sin.

In Danish, adjectives usually take the -e form after:

• a definite article
• a possessive
• certain other definite contexts

So:

• en syg datter = a sick daughter
• den syge datter = the sick daughter
• sin syge datter = her sick daughter

That is why you get syge here instead of syg.

Yes. Når hun bliver hjemme med sin syge datter is a subordinate clause.

In Danish subordinate clauses, the word order is not the normal main-clause verb-second pattern. In this sentence, you do not see a dramatic difference because there is no adverb like ikke, but the clause is still subordinate.

For example:

• main clause: Hun bliver ikke hjemme
• subordinate clause: når hun ikke bliver hjemme

So after når, the clause follows subordinate-clause word order.

Then the main clause would show normal Danish verb-second inversion:

• Når hun bliver hjemme med sin syge datter, viser hun ansvar og hensyn.

Notice:

• after the fronted subordinate clause, Danish puts the verb before the subject in the main clause
• so hun viser becomes viser hun

This is very common in Danish:

• Når jeg har tid, læser jeg.
• Hvis det regner, bliver vi hjemme.

Yes, med literally means with, but in this context it also strongly suggests staying home in order to be with / care for her sick daughter.

So the phrase can imply both:

• physical company: she is at home with her daughter
• practical responsibility: she is staying home because her daughter is sick

It is a very natural way in Danish to express that someone stays home to look after a sick child.