Since we are able to determine the enthapy of formation experimentally, why it is impossible to determine the lattice enthalpy directly by experiment?
Because you can't take an ionic solid and convert it into a gas of ions. That's not a physically realistic process. You have to infer it using a Born-Haber cycle and other quantities that can be determined experimentally.
What about theoretically calculating it from principles in electrostatics ex: as briefly noted on U Waterloo's webpage here?
For instance, for crystalline ionic compounds: Born-Lande and/or Born-Mayer equations can provide results close to experimentally determined values. The two equations both utilize the Madelung constant, which simplifies the electrostatic calculations for organized crystals. Of course these equations bring up new constants :)
(Kapustinskii took a slightly different approach to determining this.)
EDIT: In retrospect, perhaps this doesn't answer the question since it asks for experimental solutions.