I don't think taking it apart will help - when staples don't bend behind the papers, assuming the correct staples are used, it is usually either the anvil is worn or there is some misalignment between the staple and the anvil.
The anvil is usually the part that wears out first, fortunately many office staplers use a dual anvil that can be turned around to select the staples being bent outwards. This position is ideal when the staple will need to be removed at a later date.

Staples are usually produced with a sharp square end to each leg that after a few thousand staples will have done some damage to the anvil or a few hundred if the anvil had slipped through unhardened.

I suggest you take a look under strong light and magnification and compare it with that of a working stapler...