Proof that $E(\hat{\beta^{'}}\hat{\beta})=\beta^{'}\beta+\sigma^2\sum_{i=1}^k\frac{1}{\lambda_i}$

Question

I am having some difficulty proving the result above. I am considering $\hat{\beta}$ as the ols estimator of $\beta$ $(Y=X\beta +\epsilon)$ with $Var(\epsilon \mid X)=\sigma^2I$ and $\lambda_{i}$ a eigenvalue of matrix $(X'X)$.

So far, my development: $$E[\hat{\beta^{'}}\hat{\beta}]=E[[\beta+(X'X)^{-1}X'\epsilon]^{'}[\beta+(X'X)^{-1}X'\epsilon]]=E[\beta^{'}\beta+\beta^{'}(X'X)^{-1}X'\epsilon+\epsilon^{'}X(X'X)^{-1}\beta+\epsilon^{'}X(X'X)^{-1}(X'X)^{-1}X'\epsilon]$$

And now I am stuck in this part.

It would be great if someone could help me.

Thank you very much.