Two regular curves $\alpha$ and $\beta$ are considered a Mannheim pair when $N_{\beta} = \pm B_{\alpha}$ and there is a differentiable function $\lambda: I \rightarrow \mathbb{R}$ in order that $$\alpha(t) = \beta(t) + \lambda(t)N_{\beta}(t)$$
I have proven that $\lambda$ must be a constant function. Now there is this implication that I struggle with, considerd that $\beta$ is now arc-lenght parameterized: $\alpha$ and $\beta$ are a Mannheim pair if and only if $\lambda(\kappa_{\beta}^2 + \tau_{\beta}^2) = \kappa_{\beta}$.
Noting that $\lambda$ is a constant, differentiate your equation w.r.t the parameter $t$ assuming it is the arc length parameter of $\beta$ two times. Then just use the assumption you have stated, i.e. $N_{\beta} = \pm B_{\alpha}$ and take the dot product on one side w.r.t $B_{\alpha}$ and another side w.r.t $N_{\beta}$. That should give you the required condition.