Q:

Probability of finding an energy state of a non-normalisable wave-function

Suppose, say, I have the following wave function
It represents the wave function of a free particle. 
I would want to calculate the probability of finding the particle with energy ħk and energy 2ħk.   However, how do I do this? 

A:

For a free particle, energy eigenfunctions are also momentum eigenfunctions, since $\hat{H} = \frac{\hat{p}^2}{2\,m}$.
The wavefunction is a linear superposition of four orthogonal momentum eigenfunctions: two counterpropagating plane waves of wavenumber $\pm k$ and a further two of wavenumber $\pm 2\,k$, all equally weighted.  Thus there is equal probability that the energy is $(\hbar\,k)^2/(2\,m)$ and $(2\,\hbar\,k)^2/(2\,m)$.

