I don’t really understand - what is wrong with the school that you plan to attend? IMHO, I think you should repeat sophomore year now so that you fill in the gaps that you missed while you were ill. Fully commit to the school, do well in your classes, make friends, participate. Don’t even think about applying to another school as it will prevent you from enjoying and fully immersing yourself at this new school.
If you do well in your new school, you will have positive recommendations from people who will have known you for several years. You should also have extra curricular activities that represent your interests. In addition to your grades, these will strengthen your college applications.
If you persist in transferring, you should realize that this may hurt you when you apply to college. Right now, you have a story to explain your performance in freshman year and your reasons for repeating sophomore year. Moving again, will require further explanation, will make it harder to show commitment to your extra curricular activities and may make it more difficult to obtain strong recommendations.
Also, if your dad is a public figure but you still need financial aid, you’re not likely to be a development admit so your chances of admission as a junior are slim. There are schools that take non-athletic senior admits, such as Hotchkiss, but again those spots are limited.
Think carefully about why you’re desperate to get into a somewhat elite boarding school. If it’s prestige, consider that by excelling at this new school, you will have a better chance at getting into a prestigious college.