How can one reconcile the questioner's need to work (which requires intermingling with men and wearing makeup) to secure housing for herself and her family, with the rulings of Islamic Law?

Neglecting prayer is a grave sin and is impermissible regardless of circumstances or necessity. Any action that leads to its abandonment must be forsaken. As for a woman's work, it is not permissible unless it adheres to the legitimate Islamic guidelines, or there is a true compelling necessity where she finds no other means of livelihood, such as securing housing or supporting those dependent on her, knowing that owning a house is not a necessity. If the work involves prohibitions like mixing with men (unrestricted), speaking softly and alluringly to them, using perfume, or adorning herself, then it is only permissible out of a compelling necessity for which there is no alternative.