Líkamsræktin hjálpar mér að slaka á eftir vinnu.

In this sentence, líkamsræktin most naturally means exercise, working out, or sometimes the gym. The exact English translation depends on context.

The base noun is líkamsrækt, which refers to physical exercise or fitness activity. With the article attached, líkamsræktin can mean something like the exercise, the workout, or the gym, depending on what the speaker has in mind.

Because Icelandic usually adds the to the end of the noun instead of using a separate word.

So:

• líkamsrækt = exercise / fitness
• líkamsræktin = the exercise / the fitness activity

Here, -in is the definite article attached to the noun.

Hjálpar is the present tense, 3rd person singular form of the verb hjálpa, which means to help.

So:

• að hjálpa = to help
• hjálpar = helps

It is hjálpar because the subject, líkamsræktin, is singular.

Because the verb hjálpa takes the dative case in Icelandic.

So the pronoun for I / me changes like this:

• mig = me, accusative
• mér = me, dative

Since hjálpa einhverjum means help someone, and einhverjum is dative, the sentence uses mér.

Here að is the infinitive marker, like English to.

So:

• að slaka á = to relax / to unwind

After hjálpa, Icelandic commonly uses the pattern:

• hjálpa einhverjum að gera eitthvað
• help someone do something

So hjálpar mér að slaka á means helps me to relax.

Because slaka á is a fixed verb expression meaning to relax or to unwind.

The little word á belongs with the verb. It is not optional here.

For example:

• Ég slaka á. = I relax.
• Mig langar að slaka á. = I want to relax.

So in your sentence, að slaka á must stay together as one unit of meaning.

This is normal in Icelandic with many verb + particle combinations.

The verb is slaka á, but when it appears with the infinitive marker að, the usual form is:

• að slaka á

And in a main clause:

• Ég slaka á

So even though á belongs to the meaning of the verb, it often appears after the verb rather than directly attached to it.

Because eftir takes the dative case when it means after in a time expression.

The noun is:

• vinna = work

Its dative singular form is:

• vinnu

So:

• eftir vinnu = after work

Because Icelandic often uses a bare noun in everyday time expressions, just like English does in after work, before school, and similar phrases.

So:

• eftir vinnu = after work

If you said eftir vinnuna, it would sound more specific, more like after the work or after that particular work / shift.

Yes. That is also correct.

The difference is mainly:

• Líkamsrækt hjálpar mér... = Exercise helps me... in a general sense
• Líkamsræktin hjálpar mér... = The exercise / the gym helps me..., which can sound a bit more specific

In real usage, the best choice depends on context.

Yes, Icelandic word order is fairly flexible.

The version you have is a neutral, natural order:

• Líkamsræktin hjálpar mér að slaka á eftir vinnu.

But you could also say:

• Eftir vinnu hjálpar líkamsræktin mér að slaka á.

That puts more emphasis on after work. The important thing in Icelandic main clauses is that the finite verb often stays in the second position.