How do you know if the signal is a pure or mixed state when doing state reconstruction (quantum tomography)?

You can't infer the purity by observing a single axis (e.g., $hat{Z}$), so you'd indeed have to measure a non-commuting observable (e.g. $hat{X}$). Only then will you be able to gain (some) phase information which will in turn tell you about the quantum coherence (i.e., purity) of the state.

PS: I'm not sure if you can construct the whole density matrix with only two measurement axes, though. More von Neumann projections may be needed.

Doing tomography involves using several different measurement bases. If we restrict to the simplest type of measurements, projective measurements, you need at least $N+1$ of them to fully characterise an arbitrary state. To see this, note that any fixed measurement basis ${lvert u_irangle}_{i=1}^N$ pinpoints $N-1$ parameters: the probabilities $lvertlangle u_ivert rhoranglervert^2$ associated to each possible outcome. Due to the normalisation constraint, these are $N-1$ independent parameters. Therefore, if you find $N+1$ measurement bases which all give you information independent from the others, you manage to retrieve $(N+1)(N-1)=N^2-1$ independent parameters, which is the number of dimensions of the space of $N$-dimensional states.

For two measurement bases to give "independent information" in this sense, they need to be mutually unbiased bases (MUBs). The exact number of MUBs for a given dimension is still an open question, so it is not ensured, as far as I know, that you can find $N+1$ measurements that make the above argument work. Nonetheless, $N+1$ works as a lower bound: if the bases are not MUBs, then there is some redundancy in the information they provide, and therefore you need more than $N+1$ of them to get your $N^2-1$ independent parameters.

I should also note that if $N$ is the power of a prime, $N=p^M$ for some prime $p$, then we can always find $N+1$ MUBs. This is, in particular, the case for qubit systems, in which you have $N=2^m$ with $m$ the number of qubits. You can have a look at (Durt et al. 2010) for more information about this (in particular section 1.2).

Once you reconstructed the state, you know its density matrix $rho$, so you can simply compute the associated purity as $operatorname{Tr}(rho^2)$.

On a more concrete note, in the case of a qubit, $N=2$, doing tomography provides you with the expectation values $langle sigma_xrangle,langle sigma_yrangle,langle sigma_zrangle$, with which you can represent the state on the Bloch sphere, and then the state is pure if and only if sits on the border of the sphere.