Ég ætla að láta lyklasmiðinn athuga lásinn á morgun.

Because these are two different verb patterns.

• ætla að + infinitive = intend to / be going to do
  • Ég ætla að fara = I’m going to go
• láta + object + infinitive = have / make / let someone do something
  • Ég lét hann fara = I made/let him go
  • Ég ætla að láta lyklasmiðinn athuga lásinn = I’m going to have the locksmith check the lock

So:

• að is required after ætla
• athuga stays as a bare infinitive after láta

Here láta is being used in a causative sense: to have someone do something, sometimes also to make or to let, depending on context.

In this sentence, the natural English meaning is:

• have the locksmith check the lock

So the structure is:

• láta + person/object + verb

Examples:

• Ég læt hann vinna. = I make/have him work.
• Ég lét hana tala. = I let/made her speak.
• Ég ætla að láta lyklasmiðinn athuga lásinn. = I’m going to have the locksmith check the lock.

Lyklasmiðinn is the accusative singular definite form of lyklasmiður (locksmith).

Breakdown:

• lyklasmiður = a locksmith
• lyklasmiðinn = the locksmith (as the direct object here)

It appears in the accusative because it is the object of láta in the pattern:

• láta + accusative object + infinitive

So literally, Icelandic is doing something like:

• I intend to have [the locksmith] check [the lock].

Lásinn is the accusative singular definite form of lás (lock).

Why accusative? Because it is the direct object of athuga (check, inspect).

So inside the smaller clause:

• athuga lásinn = check the lock

You can think of the sentence as containing two object relationships:

1. láta lyklasmiðinn ...
   → the locksmith is the object of láta

2. athuga lásinn
   → the lock is the object of athuga

The dictionary forms are:

• lyklasmiður = locksmith
• lás = lock

The forms in the sentence are inflected:

• lyklasmiðinn = the locksmith
  
  • accusative singular definite
• lásinn = the lock
  
  • accusative singular definite

So if you looked them up in a dictionary, you would search for lyklasmiður and lás.

That -inn is the suffixed definite article, roughly equivalent to English the.

Icelandic usually attaches the definite article to the noun instead of using a separate word.

So:

• lás = a lock / lock
• lásinn = the lock

And:

• lyklasmiður = a locksmith / locksmith
• lyklasmiðinn = the locksmith

The exact ending changes depending on gender, number, and case, but here both happen to show up with -inn.

Often, yes. Ætla að is commonly used for intend to or be going to.

So:

• Ég ætla að fara = I’m going to leave / I intend to leave
• Ég ætla að láta lyklasmiðinn athuga lásinn = I’m going to have the locksmith check the lock

It usually suggests intention or plan, not just a neutral future. English going to is often the best translation, but sometimes intend to captures the feel better.

Icelandic often uses the phrase á morgun to mean tomorrow.

Literally, it looks like on tomorrow / in the morning of the next day, but you should learn it as a fixed expression meaning simply:

• tomorrow

Examples:

• Ég kem á morgun. = I’m coming tomorrow.
• Við sjáumst á morgun. = See you tomorrow.

So in your sentence, á morgun tells you when the action is planned.

Yes. Icelandic word order is fairly flexible, especially with time expressions.

These are all possible in natural Icelandic, though the emphasis changes slightly:

• Ég ætla að láta lyklasmiðinn athuga lásinn á morgun.
• Á morgun ætla ég að láta lyklasmiðinn athuga lásinn.
• Ég ætla á morgun að láta lyklasmiðinn athuga lásinn.

The version you have is very natural and neutral: the time expression comes at the end.

Yes, láta can cover several meanings in English:

• let
• make
• have

The exact nuance depends on context.

In this sentence, the normal interpretation is have the locksmith check the lock, because that fits the real-world situation: you arrange for the locksmith to do it.

If the context were different, láta could sound more forceful, like make. But here have is the most idiomatic English translation.

A useful breakdown is:

• Ég = I
• ætla að = intend to / am going to
• láta = have / make / let
• lyklasmiðinn = the locksmith (object of láta)
• athuga = check
• lásinn = the lock (object of athuga)
• á morgun = tomorrow

So the grammar is:

• subject + ætla að + láta + person/object + infinitive + object + time expression

In a more abstract pattern:

• I intend to have X do Y tomorrow

That is exactly how the Icelandic sentence is built.

Yes—this sentence is a great example of three very common Icelandic patterns:

1. ætla að + infinitive

   • Ég ætla að fara.
   • I’m going to go.

2. láta + accusative + bare infinitive

   • Ég lét hann koma.
   • I had/made him come.

3. suffixed definite article

   • lás → lásinn
   • lyklasmiður → lyklasmiðinn

If you remember those three patterns, this sentence becomes much easier to understand and reuse.