What can a professor do about an exam that was too long for the allotted time, after the students have completed it?

I found myself in a position similar to your instructor's. I considered and rejected several possibilities:

• Adjusting the curve or grading scale to compensate can be unfair to students who cannot read and understand English quickly, especially if the exam is text-heavy or the students are not native English speakers. The result of this approach is that the exam grade reflects speed of English reading comprehension more than mastery of the course material.
• Dropping the lowest exam grade, or allowing a grade on another assessment to compensate for the midterm exam grade, only works if there are other assessments that cover the material that was on the midterm exam. Otherwise, a student could get an A in the course without having mastered some of the learning objectives.
• Adding another assessment that wasn't on the syllabus, or adjusting other assessments to cover the material that was on the midterm (in violation of the syllabus), isn't fair either, especially to students who have other commitments and need to make plans in advance and schedule their time very carefully.

Ultimately, I decided that the fairest approach is to give students another opportunity (but not in the form of a required assessment) to demonstrate mastery of the course material. Depending on the course format, the size of the class, etc., several possibilities are:

• Allow students to submit a correction to their exam, and earn partial credit for answers that they didn't complete correctly on the exam but did successfully in the correction. There is a possibility that students will receive unauthorized assistance on this correction, though.
• Same as previous suggestion, but have the students explain their corrections in an oral exam. This reduces their ability to benefit from unauthorized assistance, since by questioning the students it is easier to see who really understands what they are saying. However, it is unfair to students who get nervous or have trouble expressing themselves in an oral exam.
• Grade the original exam on a curve, but allow students to take a makeup exam covering the same material, if they feel like their "curved" grade doesn't reflect their knowledge of the material. For students who take the makeup exam, the makeup exam grade replaces the curved exam grade. This may be unfair to students who have limited time to review for and take another exam, possibly because of outside commitments, and have to manage their school and other commitments very carefully.

As you can see, none of these are perfectly fair, either. Depending on the particular circumstances (and possibly with input from the class), an instructor may decide which is the least unfair.