What does the death of Ivan Ilyich teach us about happiness?

For one thing, it teaches us that happiness isn't what we often think it to be. Many of us tend to seek happiness in wealth and material goods; that's the kind of life that Ivan Ilyich has led. But it hasn't brought him happiness, because the things of this world are fleeting. It doesn't matter how good they temporarily make us feel, they can never bring us true, lasting happiness. Leading a materialistic lifestyle is a way of putting off the ultimate question—the question of our own mortality. We acquire material things and pursue endless diversions to divert attention from the question of our own inevitable demise.
It's only when Ivan Ilyich realizes the shallow superficiality of the life he's been leading that he no longer fears death as he used to. And it is then, in his final few days on earth, that he achieves the kind of true happiness previously closed off to him.