Hänglåset till förrådet gick sönder, så hon köpte ett nytt på vägen hem.

Hänglåset is the definite singular form of ett hänglås.

• ett hänglås = a padlock
• hänglåset = the padlock

In Swedish, the definite article is often added as an ending:

• en bok → boken
• ett hus → huset
• ett hänglås → hänglåset

So here, hänglåset means the padlock.

Here, till means something like for or belonging to.

So:

• hänglåset till förrådet = the padlock for the storage room
• more literally: the padlock to the storage room

This does not mean physical movement toward the storage room. It is just a common way in Swedish to connect an object with the thing it is used for.

Similar patterns:

• nyckeln till bilen = the key to/for the car
• dörren till köket = the door to the kitchen

Because it refers to a specific storage room: the storage room.

• ett förråd = a storage room / a shed / a storage space
• förrådet = the storage room

Swedish marks definiteness with the ending -et here because förråd is an ett-word.

So:

• till förråd would sound incomplete or unnatural here
• till förrådet means for the storage room / to the storage room

Gick sönder is a very common Swedish expression meaning broke, stopped working, or came apart.

It comes from:

• gå = go
• sönder = apart / broken

But you should learn gå sönder as a fixed expression. It often corresponds to English break.

Examples:

• Min mobil gick sönder. = My phone broke.
• Tvättmaskinen gick sönder. = The washing machine broke.

So in your sentence, Hänglåset ... gick sönder means The padlock ... broke.

Because gå sönder uses the verb gå, and the past tense of gå is gick.

So:

• går sönder = breaks / is breaking
• gick sönder = broke
• har gått sönder = has broken

Even though English uses break, Swedish uses gå in this expression.

Because the missing noun is hänglås, and hänglås is an ett-word.

The full phrase would be:

• ett nytt hänglås = a new padlock

When the noun is left out, the article and adjective still have to match its gender:

• en ny if the missing noun is an en-word
• ett nytt if the missing noun is an ett-word

So ett nytt here means a new one, where one = padlock.

Because the noun is already obvious from the previous clause.

English does the same:

• The old padlock broke, so she bought a new one.

Swedish often omits the repeated noun when it is clear:

• Jag tar den röda. = I’ll take the red one.
• Hon köpte ett nytt. = She bought a new one.

So ett nytt is short for ett nytt hänglås.

It means on the way home.

Breakdown:

• på vägen = on the way
• hem = home

Together:

• på vägen hem = on the way home

This is a very common expression in Swedish.

Example:

• Jag köpte mjölk på vägen hem. = I bought milk on the way home.

Because hem usually expresses direction, while hemma expresses location.

• hem = homeward / to home
• hemma = at home

Compare:

• Jag går hem. = I’m going home.
• Jag är hemma. = I’m at home.

In på vägen hem, the idea is movement toward home, so hem is the correct word.

Because så here is a coordinating conjunction, meaning so.

After a coordinating conjunction, Swedish keeps normal main clause word order:

• ..., så hon köpte ett nytt.

That means:

• hon = subject
• köpte = verb

So the order is subject + verb.

If you used a different connector such as därför (therefore), then you would get inversion:

• Hänglåset gick sönder. Därför köpte hon ett nytt.

So:

• så hon köpte = correct here
• därför köpte hon = also possible, but with a different connector

It is natural and common here because the sentence contains two main clauses:

• Hänglåset till förrådet gick sönder
• så hon köpte ett nytt på vägen hem

Swedish comma rules are sometimes a bit looser than English in everyday writing, but using a comma here is perfectly normal and helps readability.

So the comma is a good, standard choice in this sentence.