Back when I was a high school student, I had mixed feelings about compass and straightedge constructions. On the one hand, I liked the geometric challenge, on the other hand, I hated the physical challenge of working with an actual compass. Maybe 20 years later, I had exactly the same experience as a high school math teacher. I loved constructions, but I hated managing the compasses. Having to help the students whose papers tear or whose compasses' openings change even as they draw their arcs mirrors and multiplies the problems I faced when I was a student.
A partial solution is to spend less time with the physical compass, and do more on the computer. This is what I did for years when teaching high school geometry. One way to reduce physical compass use is to use other tools alongside it or instead of it, even when working on paper. For example, copy segments and angles with patty paper, or use plexiglass mirrors (Miras or GeoMirrors) for bisecting. In any case, once students master the basics, they can move on to greater challenges using dynamic geometry software — as for example in my Construction unit.
It is a problem-rich unit, starting with the traditional compass and straightedge basics, continuing with tough puzzles using GeoGebra (or Cabri, or Sketchpad,) and ending with well-known theorems about triangles. The basic philosophy is to avoid "this is how you bisect", "this is how you make a perpendicular", etc. Teaching procedures takes the life out of this subject, and does not really help mathematical understanding. Construction challenges are puzzles.
Be sure to precede the unit with this all-important kinesthetic activity.
A middle school version of the packet (shorter, possibly better worded, with more hints, and without an expectation that results will be proved formally.)
What happens when you repeatedly inscribe triangles inside each other, using the points of tangency of the inscribed circles: Triangles and Iteration, by Rachel Chou
Read my blog posts about geometric construction.
A construction approach to reviewing the triangle congruence conditions:
In my view, geometric construction is pedagogically foundational in developing a Common Core-compatible transformations-based geometry for high school. Read more about this.