What is the ruling regarding a committed girl, who does not mix with men except out of necessity, working in a mixed environment, such as a library, to alleviate the financial burden on her father?

The default is that a woman's stay in her home is better, as Allah Almighty says: "And abide in your houses" {Al-Ahzab:33}. If she needs to go out for work, there is no harm in that, provided that she adheres to the Sharia regulations, including: wearing the prescribed Islamic dress.

Working in a mixed-gender environment, its ruling varies depending on the type of mixing. If there is segregation between men and women and no cause for temptation, then there is no objection to it. This is in contrast to when the mixing includes elements that lead to temptation; in that case, it is prohibited mixing, and working in such a situation is forbidden, except out of necessity.