How do cells repair themselves

Cells repair damaged DNA, not damaged organelles. If a cell membrane is lysed or the mitochondria is severely damaged, that cell is dead.

Nope, organelles also have their own turnover and repair mechanism. However, they are very different (due to the nature of different organelles) and cannot be easily summed up in a single post. You could start looking by searching for lit for each organelle of interest and look at dynamics, turnover and repair.