Just measure in [K] basis. Since only one $|\psi_i\rangle$ among $i=1\ldots K$ can be measurement outcome, its probability $P_i=1$.
I wonder what answer is expected. Maybe you should characterize measurement in [K] basis by K projectors $|\psi_i\rangle\langle \psi_i|, i=1\ldots K$
PS: I assumed that K vectors $|\psi_i\rangle$ form a basis; this is not needed, the K vectors $|\psi_i\rangle$ may be a part of orthonormal basis.
PPS: It is not easy to explain what is asked in the textbook question because quantum measurement is very different from classical one. In classical physics, when we measure some physical value, the value always exists before measurement, and the measurement just reveals this value. Quantum measurement is very different. State before measurement may not exist at all, and we usually say about a state after measurement. But here we are said that the state before measurement exists, is one of $|\psi_i\rangle$, $i=1\ldots K$, just unknown, and with probability 1 this is the same state as we get after measurement, as measurent outcome. Hope it helps.