[susan53]

The correct sentence is :
I'm sorry that I'm late.
Here that is a subordinating conjunction. It joins two clauses (a clause is a sequence of subject, verb and (optionally) complement). So, the two clauses ...

Clause 1) I (S) am (V) sorry (C)
Clause 2) I (S) am (V) late (C)

can be joined by that : I'm sorry that I'm late

Other examples :

He told Mary that he couldn't go to the meeting.

Clause 1) He (S) told(V) Mary (C)
Clause 2) He (S) couldn't go (V) to the meeting (C)

It's important that you don't forget.

Clause 1) It (S) is (V) important (C)
Clause 2) You (S) don't forget (V)

I have an idea that we're going to play football tomorrow

Clause 1) I (S) have (V) an idea (C)
Clause 2) We (S) are going to play (V) football tomorrow (C)

As a conjunction, that is not obligatory in the sentence - it can be left out and is just "understood". Eg : I'm sorry I'm late; He told Mary he couldn't go etc. In general, the more informal the sentence is, the more likely that "that" will be omitted.

Meaning

that as conjunction is used when the second clause answers the question "What?" about the first. So :

He told Mary (What did he tell Mary?) ... that he couldn't go to the meeting.
It's important (What is important?) ... that you don't forget.
I'm sorry (What are you sorry for?) ... that I'm late.
I have an idea (What idea do you have?) ... that we're going to play football tomorrow.

So, it's not restricted to sentence types as you suggest. The things you mention like "reported speech" are just examples of sentences where clause 2 answers the question What? about clause 1.