Plot Binary Eutectic from Temperature of Melting and Enthalpy
The reason that you can't figure out how to solve for the eutectic based on the melting temperature and enthalpy of fusion of the two constituents is because, well, you can't. Those bits of information, while useful in constructing the phase diagram (well, at least the melting temperatures), are not what determines the eutectic. You will have a eutectic if and only if there are interactions between the constituents in the liquid phase that reduce the Gibbs free energy below the ideal value.

Lets make this more concrete. Take elements A and B, each with a solid and liquid phase, no intermediate phases. Taking just one phase (solid or liquid) for the moment, what happens to the Gibbs free energy when we mix them together? The first bit you get is the weighted combination of the pure phase free energies, that is:
$G_{mixture} = x_{A}G_{A} + x_{B}G_{B}$.  But is that enough? Well, no, we forgot the entropy of mixing term. So, for an ideal mixture,

$G_{ideal} = x_{A}G_{A} + x_{B}G_{B} + RT(x_{A}\ln x_{A} + x_{B} \ln x_{B})$

What does this look like? Well, taking 'fake' elements, A that melts at 950K and B that melts at 1000K, just using ideal mixtures for solid and liquid, one gets a pretty boring diagram that looks like:

with the slight separation in solidus and liquidus from the slight asymmetry in the melting points. Hmmmm... - no eutectic (since the ideal mixing impacts both liquid and solid in the same way, that is to be expected).

OK, now lets move to a regular solution model. The difference is the addition of an enthalpy term, that describes whether an A atom would rather see A or B around it (and vice versa). In the simplest way, this can be represented by a heat of mixing term using one parameter, $\Omega$. $\Omega$ is positive if the atoms don't want to mix (so, it can cause a miscibility gap), or negative if they actually prefer to mix. So, for a regular solution one gets:

$G_{regular} = x_{A}G_{A} + x_{B}G_{B} + \Omega x_A x_B + RT(x_{A}\ln x_{A} + x_{B} \ln x_{B})$

Setting $\Omega$ to -1kJ/mol, the previous diagram turns in to:



and you see that a eutectic has now formed at about $x_B = 0.25$.

Now, back to the parameters you thought were important:
The melting temperature is kind of important, in so far as an asymmetry between them shifts the eutectic of the regular solution towards the lower melting element. 

The enthalpy of fusion is not a factor at all in determining where phase boundaries are.

The interaction term(s) are highly relevant, since they determine where and how deep the eutectic is.