I think she thought the pregnancy could be saved.
Apparently she was not seeking an abortion but wanted help because her water broke 5 months early , she was bleeding and apparently she was experencing pains.
Yes they should have induced labor and delivered the fetus.
Howevre since she was only 18 weeks along the fetus would not survive deliery and apparently the Catholic Church would deem that a "direct abortion" which is against their ethics.
So they sent home twice and would have sent home a third time even though she had two inefections that could have life threatening ,but the fetus was being expeled before they could get her out of the hospital the third time.