What is the ruling on working in an educational institution that prohibits speaking Arabic and heavily favors the English language, given the presence of immodestly dressed female teachers? Is this considered cooperation in sin, necessitating immediate resignation, or is it permissible to remain until another job is found?

If you avoid mixing with immodestly dressed women, lower your gaze from looking at what is impermissible, and adhere to the Sharia (definition) guidelines, then there is no harm in you remaining in your work at the institute. You should strive to make the study of Islamic sciences and the Arabic language beloved to the students, and do not rush to see the fruits of your efforts. For your departure from work might open the door for people of evil. And your remaining there might be better than your leaving the institute, unless you fear temptation concerning your religion or falling into the forbidden.