Thanks to modern medicine, I feel much better today. I used to think it was (or "is") unnecessary to take medicine whenever I caught (or "catch") a cold. I believed that determination could (or "can") overcome any minor discomfort. However, by now I have learned to take better care of myself, and I have realized that persistence can’t solve everything. Taking medicine at the first onset of an illness not only can ease the symptoms but can actually help me to get better sooner. Since I am reasonably recovered today, I cleaned up my room and did some cooking. Organizing my life around my customary routine makes me feel more energetic and peaceful.
Don't say "the medicine" unless you mean a specific medication, like aspirin.
Your use of "endurance" doesn't sound right to me. "Endurance" is the ability, for example, to exercise for a long time. Instead, you want a word that indicates a willingness to ignore or deny signs of trouble: "persistent", "dogged", "obdurate", "strong-willed".
I think it is better to say "I am recovered today", because you ARE. That adjective describes you right now. The present perfect is not wrong, but there is no need for it.