Bad knee and back --- Best Ab workout

When my back was bad the best thing I found for me was to lay on the floor. Knees up. Hands under my butt, palms down like your grabbing the floor. And squeeze my abs in while I slowly lifted by butt off the floor. It would help me pop my back sometimes too and that was always a relief.

Probably some kind of stability exercise like planks and its variants. Most exercises (whatever exercises you can do) performed standing would indirectly target your core also as you're stabilizing the weight to not fall over, not much but it all adds up. Just be careful with your knee.
Edit: Personal favorite of mine would be the one-arm dumbbell press on a flat bench with legs up in the air, takes a lot of core to stay on the bench while performing the movement maybe give that a shot.

If flexibility is such an issue that you can't perform a "crunching" movement you'd probably want to work on that more.

Any type of core exercises. Planks (front and side, lift a leg to make it more challenging) and Super Man's (lay on your stomach with your arms stretched up over your head then lift your arms and legs of free ground, hold, relax, repeat)

The best ab workout is the one that you can do. Back injuries vary enormously.
Worth trying a few different routines to see which works for you without causing pain.

I use a gym ball ball which I find really good. Pilates, yoga, swimming, planks and core stabilty exercises all work for different people.

I don't know what they're called but my soccer team always called them italians. You lay on your back, hold the persons ankles who are standing at your head. Put your legs straight up (body @ 90 degree angle) with krnees together, and they push your legs down, not letting them hit the floor but getting close to it as you can, and bringing them back up. Do straight, left & right. Reverse crunches are killer. Where you lift up your legs instead of your head, and it works that lower "pooch". There are a lot of pilates ab exercises that are awesome.

As someone with a chronic bad back following a car accident and osteo-arthritis in both knees and one foot (yes, in my foot!), one of the best bits of advice I can come up with is that to get really good abs, you need to have strong core muscles. I had dreadful trouble with back spasms and all sorts until I discovered Pilates about 15 years ago and had some instruction right from the absolute beginner up to intermediate before stopping going to classes. I now do Pilates on my own at home because I know what I can and can't do and I also know what suits my back and what doesn't.

No I don't have good abs but that is entirely because I am obese and there is too much flab on the abs for anyone to see them! However, I do have a strong back and inner core, which means I can stand straight and tall and move well without hurting my back and also perform various exercises from that position of strength. The weight is now being tackled but I am still doing Pilates to help keep things stable.

Be careful if you are going to get instruction, to go for absolute beginner, so that you understand really properly exactly which muscles you are moving to do what exercise - yes, that sounds boring but Pilates is a long term and maintenance form of exercise, not a quick fix. If it's good enough for top sportsmen and women who took part in the Olympics, then it's good enough for me!

Try using a gym ball to take some pressure off. And have you tried kinesio tape to support your joints and back? It supports without restricting movement, and I have found it surprisingly helpful.