What will happen if mitochondria had a structure like lysosomes?
Mitochondria are double membraned organelles with an outer membrane which is a rod-like shape, and a convoluted inner membrane (Figure 1).



Figure 1

On the other hand, a lysosome is a single-membraned vesicle containing digestive enzymes, with no internal structure. Its function is to keep these dangerous enzymes in a compartment.

The "purpose" or "role" of the mitochondria is to do cellular respiration to provide energy (as ATP) to the cell. To do this, the mitochondria have many respiratory complexes (groups of proteins which make respiration happen) embedded in their internal membrane (Figure 2). As you can see in Figure 1, this membrane is convoluted with many folds; this is to increase the surface area of the membrane to have more space for respiratory complexes.


Figure 2

Moreover, these complexes pump out H+ ions into the intermembrane space. The fact that there is this thin intermembrane space allows for a buildup of H+ concentration which is crucial for ATP production (through the action of ATPase). This process is called chemiosmosis and you can learn more about it here as a starting point. A lysosome with only one membrane does not have an "intermembrane space", so this buildup in concentration cannot happen.

If you would like to learn more about mitochondria, I would recommend reading Power, Sex, Suicide: Mitochondria and the Meaning of Life by Nick Lane.