How can I calculate the energy of a solution by knowing all of it's elements?
Since you ask specifically for internal energy, would it be that you already have some ideas?

So what is internal energy? It is the energy due to the interaction among the species plus the energy due to the interaction between the species and any external field. That is, the energy it takes to "assemble" the system. 

Then for gaseous species, it would be mainly the bonding energies of the molecules. Those are usually inferred from experiments. And you can look up tables in textbooks and IUPAC website.
For ionic crystals, you would reason this way:  first ionize the species and them find the Columb interaction among them in crystal. Ionization energies are measured. Columb interaction is probably calculated and is a nightmare to calculate.
The further problem is from NaCl(s) to (aq) where the solvation involves quite a bit of interactions between the ions and water. For the case of Na+ Cl-, water molecules do not form regular cages around them, even though we CAN study the ideal case of regular cages. I think there has been some minimal models which can possibly guide a simulation study for the solvation energy. Though I don't think there has been any simulation study or any theoretical study that is sufficient to predict solvation energy.  I can be wrong. On the other hand, direct measurement should at least be able to measure the heat change during solvation.

So in digression, I don't think you can find the complete energy too well.