Did the girl commit a sin by spreading her legs during her illness and despairing of God's mercy, and is her sin great?

The girl should not have despaired of Allah's mercy, lost hope, and acted in that manner. However, since she has committed that wrong action, she should seek forgiveness and repent to Allah. Therefore, she must frequently repent and seek forgiveness, perform good deeds, and praise and thank Allah for her well-being and healing.