Is There Anything I Can Do to Fix my Crooked Smile?

We are all asymmetric if you look closely enough. The question is whether the asymmetry is enough to warrant correction, and if so what would it take to correct that.
A careful evaluation of your face would include an analysis of the muscles, but of the bone as well. We usually start with a thin cut 3-D CT scan. This shows where there are significant volume differences and significant differences in the bone. Muscle moves different on one side from another if the muscle is spannind bones with differences in projection.
Then the tough choices start about how best to achieve the maximal amount of reasonable symmetry.

You can improve the symmetry of the length of the lip with an upper lip lift and the corner of the mouth with a mid face lift with or without a tendon suspension (depends on how much muscle function you have). You may also benefit from some augmentation of the cheek and deep facial fat as this has likely atrophied from the injury, especially if you had a fracture. And finally, something like Invisalign could level the plane of the teeth to the height of the lip to help improve symmetry when you smile. 

Yes! We can use a tendon to reposition the upper lip into a more symetrical location. This would be an outpatient procedure with excellent results. 

I agree with both Dr's Aldea's and Rand's statements. Your face is asymmetric and you could achieve better symmetry by relaxing the upper lip muscles on the right side of your face. Prior to doing so, however, you should determine if you prefer symmetry with decreased upper lip movement on both sides of your upper lip. Essentially this treatment will lower your normally active right side because there are no treatments that can restore or increase activity of the muscles on the left side of the upper lip where the problem really exists.

The asymmetry with smiling is due a nerve injury weakening the muscles that elevate the lip on the injured side. To correct this, Botox can be  injected into the "stronger" side so that it can "match" the weakened side. Botox in the lower face is an "advanced technique" so find a physician that is familiar with all the facial muscles, such as a Plastic Surgeon.