Sounds like you've gone over this hundreds of times in your head with no obvious way forward. Your expectations aren't being met and all the efforts at discipline you've tried so far aren't working. If that were me, I'd be frustrated as hell!
Others have alluded to this, but in my view it is not your role to discipline someone else's children. The kids know that, which is why they aren't playing ball. Don't expect this to change - they aren't going to wake up one day and suddenly start doing what you tell them. The discipline must come from the parents, which leads me to your relationship with your partner...
By standing idly by and allowing this situation to continue, your partner is putting you in a no win situation. Having said that, I don't think he's doing it deliberately - he is caught in the middle and just doesn't know what to do.
So, after thinking about it, I would suggest 2 things:
1. Stop doing things that make you feel resentful (cooking the kids etc.). Being a martyr will only make you feel worse, and it won't make anyone appreciate you. However, be strong in your intentions - don't stop doing things for the kids in order to punish them. Rather, decide that you are not willing to do anything that further damages your relationship with them, so are not willing to build any more resentment.
2. Sit down and agree with your partner how you are going to handle the situation. You both need to genuinely agree to the point where you are willing to change your behaviour to get the result. (eg. He might have to discipline the kids more. You might have to build a better relationship with them.) I imagine that this is the area where a psychologist can be of most help to you.
Sounds like a difficult situation, and I really do wish you all the best in sorting it out.