Jeg kender ikke årsagen til forsinkelsen, men jeg ved, at bussen ikke kommer til tiden.

Here at means that, not to.

Danish at has two common jobs:

• as an infinitive marker: at komme = to come
• as a conjunction introducing a clause: jeg ved, at ... = I know that ...

In your sentence, it introduces the clause bussen ikke kommer til tiden.

Because Danish splits English know into different verbs.

• vide / ved is used for facts, information, and whole statements
  
  • Jeg ved, at bussen er forsinket = I know that the bus is delayed
• kende / kender is used for familiarity with people, places, and also certain specific nouns
  
  • Jeg kender ham = I know him
  • Jeg kender byen = I know the city
  • Jeg kender årsagen = I know the reason

So in this sentence:

• Jeg kender ikke årsagen uses kender because the object is the noun phrase årsagen
• jeg ved, at ... uses ved because what follows is a whole clause

A learner-friendly way to think of it is:

• kender
  • noun
• ved
  • clause / fact

That -en is the Danish way of marking the definite form, like English the.

So:

• en årsag = a reason

• årsagen = the reason

• en forsinkelse = a delay

• forsinkelsen = the delay

• en bus = a bus
• bussen = the bus

In Danish, the definite article is often attached to the end of the noun instead of standing as a separate word.

Because årsag normally takes the preposition til in Danish.

• årsagen til noget = the reason for something

So:

• årsagen til forsinkelsen = the reason for the delay

This is just the standard Danish pattern. Even though English uses for, Danish uses til here.

Because at bussen ikke kommer til tiden is a subordinate clause, and many Danes write a comma before that kind of clause.

You will often see:

• Jeg ved, at bussen ikke kommer til tiden.

In modern Danish, you may also see:

• Jeg ved at bussen ikke kommer til tiden.

Both can occur, depending on comma style. So the comma here is normal and very common.

Because Danish word order changes in subordinate clauses.

In a main clause, the finite verb usually comes early, and ikke comes after it:

• Bussen kommer ikke til tiden.

But after a subordinating word like at, the clause becomes subordinate, and ikke usually comes before the finite verb:

• ... at bussen ikke kommer til tiden

So this is a very important contrast:

• main clause: kommer ikke
• subordinate clause: ikke kommer

Because Danish often uses the present tense for scheduled future events, especially with transport, timetables, and plans.

So bussen kommer ikke til tiden can mean:

• the bus is not arriving on time
• the bus will not arrive on time

This is very natural in Danish, just as English can say The train leaves at 8 for a future event.

Til tiden means on time or at the scheduled time.

• Bussen kommer til tiden = The bus arrives on time

This is different from i tide, which means in time in the sense of before it is too late.

Compare:

• Bussen kom ikke til tiden = The bus did not arrive on time
• Vi nåede bussen i tide = We caught the bus in time

So in your sentence, til tiden is exactly the right expression because it is about punctuality.

Men means but and joins two main clauses.

The sentence is built like this:

• Jeg kender ikke årsagen til forsinkelsen
• men jeg ved, at bussen ikke kommer til tiden

Because men is a coordinating conjunction, the second part stays a main clause, so jeg ved has normal main-clause word order.

That is why you get:

• men jeg ved ...

and not a special subordinate-clause pattern after men.

Yes. That is also correct and very natural.

It means essentially the same thing, but the structure is different:

• Jeg kender ikke årsagen til forsinkelsen
  literally: I do not know the reason for the delay
• Jeg ved ikke, hvad årsagen til forsinkelsen er
  literally: I do not know what the reason for the delay is

So the original sentence is fine, but this alternative is also common Danish.