This problem is related to the absence of symmetry breaking in the finite-size quantum Ising model. In an infinite system, the ground state is twice degenerate if $\lambda > 1$. Therefore, states with broken symmetry $\hat{\sigma}_i^x\rightarrow -\hat{\sigma}_i^x$ are possible for which $\langle\hat{\sigma}_i^x\rangle\neq 0$. For any system of finite size, the ground state is non-degenerate, so it must be symmetric with respect to the transformation $\hat{\sigma}_i^x\rightarrow -\hat{\sigma} _i^x$, and it is impossible to have $\langle\hat{\sigma}_i^x\rangle\neq 0$. An indication of symmetry breaking at $N =\infty$ in finite-size systems is the exponential smallness of the energy gap, $E_1 - E_0\sim O(e^{-Nc(\lambda)})$ if $\lambda > 1$.
So, if $\lambda > 1$, I would advise you to perform calculations for the modified Hamiltonian $$ \hat{H}_1 = \hat{H} - h_1 \sum_{i = 1}^N\hat{\sigma}^x_i, $$ with the parameter $h_1$ satisfying the conditions $E_1 - E_0 \ll h_1 \ll E_2 - E_0$.
A short but fairly complete discussion of symmetry breaking in the quantum Ising model can be found in he first chapter of the Franchini's book