How long does it typically take for a injured knee ligament to heal?

Ligament dependent. Not all knee ligament injuries can be treated the same. Most mild sprains can be managed nonoperatively and heal on the order of weeks to a month or two. Physical therapy and bracing may be indicated to optimize recuperation. Complete tears or grade iii sprains may require surgical intervention (acl, lcl, some PCL and few mcl grade iii injuries), thus requiring much more time for healing/recovery.
One week to never. This question can't be answered in a few short sentences because the answer depends on which of the several ligaments about the knee is/are injured and how severely the ligament(s) is/are injured. For example, a mild sprain of the medial collateral ligament typically heals within a week or two whereas a torn anterior cruciate ligament rarely heals often requiring surgery to make a new ligament.