Is it permissible to complete studies in a mixed, non-Muslim school for five months, given that no non-mixed, Islamic alternative is available, and that the mother will be affected by leaving this school?

The basic ruling is that it is impermissible to study in mixed-gender schools due to the corruption that arises from them. However, if there is a pressing necessity for it and no other schools are available, then there is no harm in continuing one's studies, provided one commits to lowering the gaze, staying away from causes of temptation, and being mindful of Allah. It is advised to be diligent in fortifying oneself, avoiding allurements, and refraining from attendance if one feels oneself drifting towards the forbidden.