How can I repair a completely torn knee ligament injury without surgery?

Cannot be done. Ligament is a tough structure that is beneath the skin and currently we don't have the ability to repair a torn ligament without surgery. We can do minmially invasive laparoscopic surgery, but still considered surgery because it requires cutting/piercing holes to insert the scopes--see picture here. But you may not need surgery at all if ligament is "not essential." consult doc good luck.
Depends on Ligament. The knee has 4 main ligaments. The mcl, and lcl usually heal on their own. The acl and PCL do not. They usually need surgery. Pcl tears that isolated do not usually need surgery depending on symptoms. Acl tears usually need surgery, unless the patient is a sedentary, older individual, or if the patient is a coper.