Keðjan á hjólinu er orðin of laus, svo við förum með það á verkstæði á morgun.

Because Icelandic usually adds the definite article onto the noun instead of using a separate word like English the.

• keðja = chain
• keðjan = the chain

So Keðjan á hjólinu means the chain on the bike.

Because á here means on in a location sense, and when á expresses location, it normally takes the dative case.

• hjól = bike / wheel
• hjólið = the bike (nominative/accusative)
• hjólinu = the bike (dative)

So:

• á hjólinu = on the bike

er orðin means has become or has gotten.

It comes from the verb verða = to become.
In this sentence:

• er = is
• orðin = a participle meaning become/gotten

Together, er orðin expresses a change of state:

• Keðjan er orðin of laus = The chain has become too loose

This is a very common Icelandic pattern.

Because orðin agrees with keðjan, which is:

• feminine
• singular
• nominative

The participle orðinn changes form to match the noun it describes.

So here:

• masculine: orðinn
• feminine: orðin
• neuter: orðið

Since keðjan is feminine, Icelandic uses orðin.

of means too or overly.

So:

• of laus = too loose

This is an adverb placed before an adjective.
It is not related to the English word of.

Other examples:

• of stór = too big
• of dýr = too expensive

Because laus is the correct form of the adjective here, even though the noun keðjan is feminine.

Icelandic adjectives agree with the noun they describe, but the feminine singular form does not always end in -a. In this case, the feminine nominative singular is simply laus.

So:

• keðjan er laus = the chain is loose

That may feel surprising at first, but it is just how this adjective declines.

Because Icelandic often uses the present tense to talk about the future when the time is clear from context.

So:

• við förum ... á morgun = we’re going ... tomorrow / we’ll go ... tomorrow

This works much like English:

• We leave tomorrow
• We’re taking it tomorrow

The phrase á morgun already tells you the action is in the future.

Here fara með eitthvað means to take something or bring something somewhere.

So:

• við förum með það á verkstæði = we take it to a workshop

Literally, it looks a bit like go with it, but in context it means take it along / bring it.

This is a useful expression to learn as a whole phrase:

• fara með bílinn á verkstæði = take the car to a workshop
• fara með hjólið í viðgerð = take the bike in for repairs

It most naturally refers to hjólið — the bike, not the chain.

That makes sense from the context: if the chain is too loose, you would usually take the bike to the workshop, not just the chain.

Also, hjól is a neuter noun, so the pronoun is það.

If the sentence were referring to keðjan instead, you would expect a feminine pronoun instead.

Because Icelandic often uses á with certain destinations, especially places thought of as institutions, workplaces, or functional destinations.

So á verkstæði means:

• to a workshop
• to the repair shop / garage

Also, with movement toward a place, á usually takes the accusative. In this noun, though, the form verkstæði looks the same, so you do not see a visible change.

So the phrase is best learned as a natural expression:

• fara á verkstæði = go to a workshop / garage

Because verkstæði is indefinite here:

• á verkstæði = to a workshop
• á verkstæðið would be to the workshop

The sentence does not specify a particular workshop, just a workshop.

Again, Icelandic usually adds definiteness to the end of the noun:

• verkstæði = a workshop
• verkstæðið = the workshop

svo here means so.

It connects the two parts of the sentence:

• The chain on the bike has become too loose, so we’re taking it to a workshop tomorrow.

It is a very common linking word in Icelandic.

Yes. á morgun is the standard Icelandic expression for tomorrow.

Even though it literally contains á, you should mostly learn it as a fixed time expression:

• í dag = today
• á morgun = tomorrow
• í gær = yesterday

So in this sentence:

• á morgun = tomorrow