I was ill for a couple of days and I could not observe the prayers due to a severe illness, even when I tried to observe with either my eyes or heart. Besides, I feared that I was unclean, and each time I wanted to pray, the drugs they gave me made me feel dizzy and at times, I fell asleep. Now that I am ok, how do I make up the missed prayers?

You are not required to make up any of the obligatory prayers if you were unconscious during the entire time period of the Salah. For instance, if you fell unconscious from 10 am to 5 pm, you do not have to make up the thuhur (noon) prayer because you were incapable (and therefore unaccountable) during that time. As for the other prayer times, during which you were awake, you must do your best to fulfill all the pillars required for the correctness of your Salah, and then perform your Salah on time. No excuse is acceptable for a Muslim not to pray, even if he does not have Wudu, or cannot face the Qiblah, etc.; he should pray as is. And if what he did was the best he can ever do, then he does not even have to make these prayers up.