You do not seem to be anywhere close to understanding the difference between inoculation, which is what Washington did, and vaccination, which had not yet been discovered.
Both pertain to certain kinds of diseases, that you can usually only get once. Once you've had the disease, your immune system knows how to stop that particular pathogen from affecting you again.
Vaccination, which had not yet been discovered at the time, involves using a weakened dead, or denatured form of the pathogen; you don't get sick from it, but it shows your immune system how to fight that pathogen, so that later, if you are infected with that pathogen, the immune system can fight it off right away.
Inoculation is what happened under Washington. This involves exposing the patient to the full pathogen. The patient does get sick, experiences the full symptoms and course of the disease, but after the disease has run its course, the patient is thereafter immune to getting that disease again. Those inoculated with smallpox got sick with smallpox; experienced all the symptoms, all the full course of the disease. The idea was to get the disease over with at that time, so that at a later time, when they were needed to fight, they would not be vulnerable then to getting the disease.