Jeg samler glasskåret op med det samme.

Because Danish usually puts the definite article at the end of the noun instead of using a separate word.

• et glasskår = a shard of glass
• glasskåret = the shard of glass

So -et here is doing the job that the does in English.

A separate definite word such as det is used in some other cases, especially when there is an adjective:

• glasskåret = the shard of glass
• det store glasskår = the big shard of glass

Glasskåret is the definite singular form of et glasskår.

The forms are:

• et glasskår = a shard/piece of glass
• glasskåret = the shard/piece of glass
• glasskår = shards/pieces of glass
• glasskårene = the shards/pieces of glass

So the sentence as written suggests one specific shard/piece.

If the intended meaning is more like the broken glass in the sense of several bits of glass on the floor, Danish would often more naturally use glasskårene:

• Jeg samler glasskårene op med det samme.

Because samle op is a particle verb, much like English pick up.

So:

• samle op = to pick up

In a main clause, Danish often places the finite verb early in the sentence, while the particle op comes later:

• Jeg samler glasskåret op.

This is very similar to English:

• I pick the shard up.

Even though samler and op are separated in the sentence, they still belong together in meaning.

That is the normal word order with this kind of verb in a main clause.

A simple way to see it is:

• Jeg = subject
• samler = finite verb
• glasskåret = object
• op = particle

So the pattern is:

subject + verb + object + particle

Examples:

• Jeg tager bogen op.
• Hun skriver nummeret ned.
• Vi rydder det op.

With samle op, this order is completely normal:

• Jeg samler glasskåret op.

Because Danish very often uses the present tense to talk about the near future, especially when the context makes the timing clear.

Here, med det samme already tells you that the action will happen right away, so the present tense sounds natural:

• Jeg samler glasskåret op med det samme.

This can mean something like:

• I’ll pick it up right away
• I’m picking it up right now

If you want to sound more explicitly future-like, Danish can also use skal:

• Jeg skal samle glasskåret op med det samme.

But the simple present is very common and natural.

Med det samme is a fixed expression meaning:

• right away
• immediately
• at once

It is best learned as a whole phrase, not word by word.

So even though it literally looks like with the same, that is not how it works in actual meaning here. It is simply an idiomatic expression for immediacy.

Other Danish words with a similar meaning include:

• straks = immediately
• øjeblikkeligt = immediately, instantly

But med det samme is very common in everyday speech.

Yes, it is natural if you mean one specific piece of glass.

But there is an important nuance:

• glasskåret = one shard/piece
• glasskårene = several shards/pieces

So if the situation is that a glass has shattered and you mean the broken glass as a whole mess on the floor, many speakers would more naturally say:

• Jeg samler glasskårene op med det samme.

If the object is already understood from context, Danish might also simply use a pronoun:

• Jeg samler det op med det samme.

So the original sentence is fine, but it sounds most natural when talking about one particular shard.

The sentence follows normal Danish main-clause word order:

• Jeg = subject
• samler = finite verb
• glasskåret = object
• op = particle
• med det samme = time expression/adverbial

So the structure is:

subject + verb + object + particle + adverbial

Danish main clauses are generally verb-second languages, which means the finite verb likes to stay early in the sentence.

For example, if you move med det samme to the front, the verb still stays in second position:

• Med det samme samler jeg glasskåret op.

That word order rule is one of the most important patterns in Danish.