Absolutely don't focus on trying to get a knee down!
In 30+ years of riding on the street, I have never touched a knee down. Shift weight around on the bike - standing on the pegs and shifting about - Absolutely. But not knee down. It requires cutting into safety margins too far and taking chances with uncertain pavement conditions and traffic that I'm not willing to take on the street.
On the racetrack ... it just happens. Again, it's not an objective to try to do it. It just happens.
So now, try this. Find out if your steering is "neutral". When you are mid-corner after you have set your lean angle and cornering trajectory, the bike should require no steering input on the bars in order to maintain that trajectory. It should feel like you could take your hands off the bars and it would continue its path - you could steer it with your fingertips. That's what it should feel like - "neutral".
The situation of the bike wanting to tighten its line, or continue falling into the corner, is not common (the other direction is a lot more common) but it can happen. Possible causes:
- Tires. Pressure too high, or incorrect tire sizes for the rims, or tire profiles too "pointy" for what the geometry of the bike wants, whether due to high wear on the edges (ex race tires) or simply the nature of the tires that were chosen.
- Geometry. This relates to the steering head angle and the amount of trail in the geometry (these are connected) - these are related to whether the bike has a nose-down tail-up pitch attitude (which heads in the nervous/twitchy/unstable direction) or a nose-up tail-squatting attitude (which heads in the heavy/stable/"riding through glue"/"steers like a barge" direction), and both of those relate to the rider-aboard ride heights front and rear ... too low in the front, or too high in the rear. Ride height and preload (and spring rate!) affect the pitch of the bike when it settles down in the corner. I found out last year with my race bike that the rear spring rate has a very strong influence on this!
- On the subject of geometry ... a bike that has been crashed (frontal impact) and has a bent frame that has the effect of shortening the wheelbase ... also has the effect of steepening the steering head and shortening the trail ... unstable ... see above.
- Rider behaviour. Hanging off too much for the amount of lean angle being requested ...
- Weight distribution, both fore/aft and vertically. BUT ... I am operating on the assumption that this is a sport bike and the rider in question is riding solo (not carrying a passenger), weighs somewhere in the vicinity of what the bike's suspension is designed for, and does not have a way-up-high-and-back luggage rack installed.
I am operating on the assumption that the bike in question is the one in the original poster's signature - a late model Honda CBR600RR. I've ridden those, on the street, and on the racetrack. There is nothing "wrong" with those, the way they came out of the factory. The steering is biased towards being rock-solid stable, not nervous. (If you think a CBR600RR is nervous, don't ride an R6. Personally I want the bike to turn, and I'll deal with the stability or lack thereof later.)
