When do we useinversions in Reported Speech?
I mean if we have indirect objects and different attributes, adverbs and so on?
You don't use inversions in reported speech, as far as I have ever seen.

Can you give a few examples and say what you think needs to be inverted?
Students: Are you brave enough to let our tutors analyse your pronunciation?
" I don't like the cake!" Jimmi exclaimed \ exclaimed Jimmi
Yet " I don't like the cake", Jimmi said abominably.
OK. Now I understand what you mean. Emotion: smile
These are in free variation when you are writing dialog in a narrative. You can invert some and leave others uninverted, just for variety. In either case, leave the adverbs at the end.
"...." said Robert with a smile.
"...." Robert said with a smile.
"...." shouted Lucy impatiently.
"..." Lucy shouted impatiently.