Hun tørrer sveden af med et håndklæde og lægger hængelåsen tilbage i tasken.

In Danish, the ordinary present tense often covers both:

• a general action: she wipes / she puts
• an action happening right now: she is wiping / she is putting

So Hun tørrer ... og lægger ... is completely natural for an action in progress.

If Danish wants to strongly emphasize that something is going on right now, it can use other expressions, such as er ved at, but that is not necessary here.

Tørre ... af is a very common Danish particle verb. Here af adds the idea of off / away.

So:

• tørre = dry, wipe
• tørre ... af = wipe off, wipe away

The object often comes between the verb and the particle:

• Hun tørrer sveden af
• compare English: She wipes the sweat off

So the split is normal. Danish does this with many verbs:

• tage skoene af = take the shoes off
• skrive det ned = write it down

Without af, tørre can still make sense, but af makes the idea of removing something much clearer.

That -en is the usual Danish way of marking the definite singular for many nouns.

So:

• sved = sweat

• sveden = the sweat

• hængelås = padlock

• hængelåsen = the padlock

• taske = bag
• tasken = the bag

Unlike English, Danish usually adds the definite article to the end of the noun instead of putting a separate word before it.

These nouns are definite because they refer to specific things understood from the situation: the sweat on her body, the padlock she has, the bag in question.

Also, Danish often uses the definite form where English might use a possessive:

• English: her bag
• Danish very often: tasken, if it is obvious whose bag it is

Because håndklæde is a neuter noun in Danish, so it takes et in the singular indefinite form.

• et håndklæde = a towel

Danish nouns are mainly either:

• en-words (common gender)
• et-words (neuter)

So you simply have to learn which article belongs to each noun.

Useful forms here are:

• et håndklæde = a towel
• håndklædet = the towel

Notice that the definite ending for a neuter noun is usually -et, not -en.

Because the same subject, hun, applies to both verbs.

So the sentence structure is basically:

• Hun tørrer ... og lægger ...

This is like English:

• She wipes ... and puts ...

You can think of it as one subject with two coordinated actions.

Danish can repeat the subject if needed:

• Hun tørrer sveden af med et håndklæde, og hun lægger hængelåsen tilbage i tasken.

But without the second hun is more natural and compact when it is clearly the same person doing both actions.

This is an important Danish distinction:

• lægge = to lay / put something somewhere
• ligge = to lie / be lying somewhere

So:

• Hun lægger hængelåsen i tasken = She puts the padlock in the bag.
• Hængelåsen ligger i tasken = The padlock is lying in the bag.

In your sentence, she is actively moving the padlock, so Danish uses lægger.

This lægge / ligge contrast is similar to English lay / lie, although Danish uses it very consistently and learners need to pay attention to it.

Tilbage means back. So:

• lægger hængelåsen tilbage i tasken = puts the padlock back in the bag

It suggests that the padlock was in the bag before, or that the bag is its usual place.

As for the preposition:

• i tasken = in the bag / into the bag

With a verb of placement like lægge, Danish often uses i where English would naturally say into. The movement is already clear from the verb lægger.

So:

• Hun lægger den i tasken = She puts it in/into the bag.

Why not til?

• til tasken would mean something like to the bag or toward the bag, not inside the bag

Why not ind i?

• ind i tasken is possible, but it adds more emphasis to the motion inward
• i tasken is the normal neutral choice here

And again, tasken rather than sin taske is natural because the bag is already understood from context.