How do I make a face? I thought it was the F key, but that doesn’t work.
SHIFT+F will cover your hole… but it wont obligatory be on squares. 
Also, should your contour is NOT lying in a flat surface, dont expect the cover to follow the surface before deletion, i.e. you may need re-alignment.
Regards,
Well Blender753 that won’t make a face with what you have done. You have to do it one triangle at a time and then when that is done just extrude one verticie and fill that with the triangle. When you then have two triangles creating a square go into face select and select the two triangles. Press f and this will fill them to transform it to a square
BlendFace
Actually is there a quicker way in doing what is said? Could you just extrude and scale?