Which is better: for a sick woman to fast despite her fatigue, or to break her fast so she can be strong enough for other acts of worship?

If fasting increases your illness or delays your recovery, then you are permitted to break your fast. Making up the days later is preferable, even if it is in winter, as Allah Almighty says: "And whoever is ill or on a journey - then an equal number of other days." However, if your illness is incurable and you are unable to fast or make up the days, then you must feed a poor person for each day you break your fast, as Allah Almighty says: "And for those who can fast with difficulty, a ransom is the feeding of a poor person."