1.A serious illness (did).
2.Difficulties/Problems/Troubles/Challenges.
3.She swam a ten-mile marathon and set women's record.
Or:She swam a ten-mile marathon,setting the women's record.
Or:She set the women's record for a ten-mile marathon swimming.
4.Because she had tried her best (to fulfill the task).
Or:Her strength of purpose was as important as reaching Cuba.
5.I am encouraged by her determination and strong will.
Or:I am inspired by her courage and strong desire to succeed.
Or:From her I learned that once we set a goal,we should try hard to achieve it.
Or:We should never give up in the face of difficulty/failure.
Or:We should never stop trying new things and we may achieve success in different fields.