No weights?
Try:
Single Leg Squats (as deep as you can)
Single Leg or bulgarian Squat Jumps
Scissor Lunges

Good exercises that you won't be able to do tens and tens of reps of (if it gets to a point were you can do loads and loads of a movement then you are not having to generate enough force to stimulate strength development, so you either need to move more forcefully and explosively or add extra resistance in the form of a weight, seeing as you have no weight these should do and because they are dynamic you should develop good knee stability... cross trainer and the bike are not giving enough resistance and they are 100% stable so your central nervous system doesn't have to develop to improve gain stability)

To be honest it doesn't sound to me like your knee joint (or the muscles bridging it) are weak as such, I'd guess that they are over active and tight (espeicially if you cycle/cross train a lot) and this is what is causing the clicking. Stretch them and get some foam rolling done to loosen them and start training your glutes to function better (most people have tight quads and bad knees because their glutes don't work)

Try practice contracting your glutes by doing back bridges and learn to squat with correct form (big chest, arse out). Best way to squat properly is to do it facing a wall with your toes touching the wall. When youe squat, your knees shouldn't pass your toes... if they do it's a sign of weak/inactive glutes and that will be what is fucking your knees up most likely.