Do I have to re-make up the prayers for the days of an illness that afflicted me, during which I was semi-unconscious and my prayers were not performed properly, or is what I performed sufficient based on my family's observation?

If one loses their mind for a certain period, prayer and fasting are not obligatory during that time, because the intellect is the basis of accountability (مناط التكليف). There is no obligation to make up the missed prayers and fasts, unless the questioner regains consciousness during the time of a prayer; in which case, that prayer becomes obligatory for her, along with any prayer that combines with it before its time.