About a year ago I suffered a major knee injury and while I have been cleared to return to the dojo and train, I still have some weakness in the knee. What are some good exercises to strengthen the knees for when you are in low frontal stances such as shiko dachi (四股立)?
As mentioned in this and other answers to that question one of the biggest things you can do to help your knees are progressively more difficult bodyweight squat variations or barbell/dumbbell squats (preferably with help of a trainer). These have a direct impact on the muscles that are responsible for stabilizing the knee, and done correctly they are tremendously valuable for these sorts of things.
The big thing is to make sure your form is correct and that you are going below parallel (if your form isn't correct they are hell on your knees), and I'd talk to your physical therapist about any particular considerations, if squats are counterindicated by your particular condition, and if so what alternatives exist that would serve a similar purpose. I wear a knee brace when doing squats on recommendation of my chiropractor (who is also a physical therapist).
The other thing, as mentioned by stslavik, is to do the stance or as closely as you can reasonably manage. My experience here is that "lots of light practice" is a good way to go: We used to advocate rapier students go into their fighting stance and hold it during commercial breaks while watching TV, or just for a few minutes here or there throughout their day. So rather than trying to do it only in class, on days when you don't have class take a few minutes here and there throughout the day to practice. Sink into it (or as close as you can manage without pain), hold it for a little while (letting up if you can't manage it, no sense further injuring yourself), maybe do a little of the footwork from that stance or do a little practice from that stance, and then go on with whatever you are doing. Listen to your knees and don't overdo it, but keep practice to whatever degree you can.
Not to sound terribly flippant or dismissive, but the best way to build up to doing a stance is to do the stance.
When we perform a specific activity, we engage the muscles necessary for that activity. If you want, for instance, to effectively chop wood, then you should chop wood. This also serves the secondary purpose of building up your neural pathways to reacquaint your body with what is potentially going to become a new way of performing that action; if you have any sort of nerve damage or muscular damage, the surrounding muscles and tendons will compensate to the best of their ability to protect the damaged region as well as the body as a whole.
Unfortunately, physical therapy only goes so far – their interest is largely to get you back up to the ability to live an ordinary life. Your life, as a martial artist, is extraordinary, and thus has higher demands of your body.
The real key is to perform the activities you need to perform, but slowly and carefully building up to deeper and more intense variations of that activity.
I have a knee injury that I take care of by an excersize recomended by my orthopedic surgen. It's basically knee bend while hovering and moving your other leg around your body in different positions. Here you can see an instructional video of how to do it. The exercise will strengthen the muscles around the knee and the ankle and I've found that it helps me hold my forms better.
In agreement with @stslavik, doing the stance and using the stances during your training will be the most beneficial to your ankle/calf/knee/quads/hips/lower back strength.
Pay careful attention to remaining in the stance during the technique, and when moving through the kata. Take care to keep your feet correctly aligned (according to the style of your training). If you're supposed to stay low, then remain low all the way through - i.e. don't "pop up/stand up" with a straightened leg.
If you're trying to improve leg strength then worry less about getting the technique to work and concentrate on doing it right.
On a knee injury, you should concentrate on improving mobility and strength in the knee "joint" muscles, and in your lower back. Squats (slow and low, back straight) are great.
If you are in a stance and your knee hurts (glancing pain), then most likely your weight is too far forward. If you put more weight in your heel, you'll find that your quads get to work much more. When your knees begins to hurt that way, ease up from the stance. Eventually the knees will get stronger and the type of 'pain' you get will change.
Now - it is all well and good to be able to get into a stance without hurting your knees, but the problem is, a stance is only good if you can move with it. It's important to remember that the knee is only a bidirectional joint. You can bend the knee and you can straighten the knee, but if you attempt to turn the knee sideways, you'll soon realize, probably painfully, that the body is not meant to do that. So, you're down to two options: the ankles and the hips. The ankles are useful for the smaller tweaks, mostly the ones involving the feet (which I'm sure you already knew, even if you weren't aware of it). This only leaves one option - the hips. For proper mobility, it is CRITICAL to have good hip flexibility. The hip is a rotating joint, so it can move your leg. Even better, you can move BOTH legs! Now, if you attempt to move in a way which the hip cannot accommodate, the movement will stop. If you force it, other joints will try to accommodate you.. And the knee will try, and then possibly give out, if you ask for a movement which SHOULD be a hip movement and turns out to be a knee twist.
The short of it is, don't do that. :)
It depends on the injury!
If it's primarily a muscle injury, you can do well with squats, leg strengthening, 1 legged balancing, and of course, the stance itself.
If you've had a tendon damaged (replaced, torn, restapled to bone), then the issue is that the joint itself is unstable. Again, you'll want to do strengthening leg exercises but at lighter weights and slower. If you can do single legged balancing, this can help (hold on to a wall if you need to initially). Slow leg kicks and similar control focused exercises will also help build the muscles up and the muscle intelligence to keep the stabilization of the leg.
When that's strong, try using a balance board as well.
If you've had cartilage damage - meniscus discs, etc. I'd be very careful and honestly, you may not be able to do some stances or deep stances. Cartilage barely repairs if at all, and isn't really replaceable. You may end up having to simply forego some stances and adapt your movement/fighting style. More pressure on damaged cartilage doesn't heal or toughen it - it continues to wear it away.
The general rule is that muscle strengthens with use, tendon can be toughened over long periods of time, joints can be protected with strong, reactive muscles, but cartilage has to be preserved. A lot of schools/teachers/training often will treat all types of injuries as something to "tough out and work through" which can make things worse! Be careful with your body, train right so you can get the most out of it.
If your injury is a combination of the above 3 (common for sudden injury, falls, car accidents, etc.) then use the worst case scenario as the protocol, and be kind to yourself. Whatever exercises the PT has given you, KEEP DOING THEM.