K^+ can not exists because electrons are removed from an inner shell. The ion O^- can not exist because it will have an incomplete octet.
Stability of ions
Ions are species that contain a positive or a negative charge. When we consider the electron configuration of ions, we can deduce the stability of the ion.
Let us consider the electron configuration of K^2+ which is 1s2 2s2 2p6 3s2 3p5 which means that the second electron is to be removed from an inner shell which implies greater energy expenditure hence it does not exist.
In the case of O^-, involves the formation of an ion with an incomplete octet which is not possible.
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