Replace the highlighted part: She insisted me to take the medicine.

Explanation

The verb insist is not followed by "insist someone to do something."
The correct structure is:
insist that + subject + base verb (or should + verb)
So the correct sentence is:
"She insisted that I take the medicine."
that I take the medicine – Correct
me that I take medicine – Incorrect structure
me taking the medicine – Incorrect structure
to take medicine by me – Incorrect and unnatural
Correct Answer: that I take the medicine