I like much about Kierkegard and his writings, but I find myself thinking of despair in its simpler context as an absence of hope. But this is just more quibbling over definitions. The fact is our language is rich and has many words for such sorrowful emotions as are here mentioned, with definitions that are approximate and overlap, while labeling emotions that are far from distinct in nature. Who shall say where despair ends and depression begins when even our psychiatrists and psychologists have conflicted understandings about them?
I have known a number of people who suffer from depression or manic-depression, and even in the same person, the manifestation of the illness can vary considerably. Sometimes one sees self-hate, sometimes emotional numbness, sometimes desolation and despair, sometimes even a visible rage. These feelings can appear quite irrational since the onset of such depressive events are caused more by chemical events within the brain than by reaction to experiences.
Despair is less clinical in definition. We often speak of people feeling despair when they suffer great disappointments in their lives, or when people live in an ongoing condition of grief or hopelessness. We are not here talking about biochemical states of the mind. It is more likely a reaction to experiences.
But is it a proper reaction to experiences? Well, I guess you have to walk a way in those shoes to know the answer to that one.