What are some broadening assignments for E5 that are KD complete?

I think you have some of these terms confused. An assignment is not something that you take to fill time between now and your next promotion. Your primary broadening assignments are drill sergeant and recruiter, and possibly instructor if you are jtac certified.
What you should be focused on right now for career progression is Advanced schooling. Ranger school, jtac, airborne, air assault, jumpmaster, Pathfinder, combatives, battlestaff, fries master, slic, or anything else like that

I don't think KD means what you think it means brother... How do you figure you're KD complete?

If you completed your operational assignments. Volunteer to be DS.