How can we tell the particle size of solutes in solutions?
I assure you there are many reasons to believe that the ions really are solvated individually. The first thing to look at is the relative strength of interactions between the ions in a lattice and the ions when in a solution. For water, these interactions must be very strong because a number of hydrogen bonds must be broken, the interaction between ion and water must overcome this initial deficit and the loss of some entropy. So, the interaction is surely quite strong.

It is of course possible for two ions which dissolve to get back together and interact with each other. They are oppositely charged ions after all. By looking at the relative size of equilibrium constants for these two processes, however, I think you'll be convinced it is far preferable for the ions to be solvated. For instance, a quick google search will tell you that the $K_{sp}$ of $\ce{NaCl}$ is around $36$. That's an astronomically large solubility product. This means we dump sodium chloride into water until the concentration of both ions is around $6\ M$.

Thus, the easiest experiment to do is to just saturate a solution of water with salt and write the mass of salt added in molarity units. This doesn't really answer your question though because we assumed that the salt existed as two ions and then made a correspondence between that concentration of ions and the concentration of water which each ion is occupying. Plus, we would generally be interested in more dilute solutions than this.

This is why people have used methods which probe the structure of ions in a solution directly. Namely, using x-ray diffraction and neutron scattering experiments, it is possible to determine the local environment around water (or something else solvating an ion) and calculate the hydration number. This is the number of ions, of a specific ion, solvated per molecule of solvent.

Here is a review which discusses x-ray diffraction and neutron scattering experiments and one which discusses coordination numbers of ions solvated in water.

Using these structural methods, which show it is extremely unlikely there are chunks floating around in solution, the following data can be found for hydration numbers of ions in water:
$$n_{\ce{Li+}}=4$$
$$n_{\ce{Na+}}=5$$
$$n_{\ce{K+}}=6$$

Here is also an old review which describes many simple but outdated methods for determining the solvation/hydration numbers of ions in a solvent.

Hopefully some of this is helpful and convincing. The experiments are quite definitive, but if you believe the equilibrium constants, then it is very hard to believe these lattices would remain intact in solution.