STEM Mentoring and Inspiration Act of 2014 - Amends the Immigration and Nationality Act to add science, technology, engineering, and math (STEM) engagement programs that incorporate self-directed student learning, inquiry-based learning, cooperative learning in small groups, collaboration with mentors in the field of study, and participation in related competitions to the National Science Foundation (NSF) competitive grant program for K-12 math, science, engineering, and technology education.