How to get my flexibility back after a broken elbow?

Therapy. Directed therapy by a certified occupational therapist is the best way to get motion back. If therapy fails after all treatment options are exhausted then surgery may be needed. Usually the motion can be regained without surgery. It also depends on the severity of the fracture and if surgery was originally needed to fix it.
Therapy. The only way to get the maximal amount of recovery is to perform aggressive therapy after your doctor clears you to begin. Therapy is typically 2-3 days a week but you should be doing some therapy at home as well to maximize your recovery.