If she doesn't finish her chores then DH has to do them for her. His responsibility. If he lets her off of doing them to watch TV with him then he assumes the responsibility for doing them himself ... as her parent. As for the homework? Not your problem. Ket DH deal with the teacher who complains or who gives his daughter a fail for not being up to scratch or turning in all her homework. You put everything back on his shoulders ... where it belongs.

Exactly what Wintwrglow said. Dishes aren't done, make DH take care of it. SK needs something, DH takes care of it. Its hard at first, but totally worth it. You get to sit back and watch as in time DH starts to get annoyed with SDs behavior once he is dealing with it on his own and you are no longer being the buffer.

Pensive Panda, I FEEL for you SO much! Having been in the exact same situation myself, I can tell you, it IS absolutely maddening, when you're just trying to do the right thing by these kids. Personally, I have 5 of my own, as well as a 13 SD who is EXACTLY the same as yours. My 5 are grown, and thankfully all doing well, because I don't have the energy left after my 13 yr old SD. It is almost impossible to fully disengage when having them full time,(this is my second marriage after 20 years with 3 Srepkids from my 1st),. I just did the best I could for my own sanity. It is in my opinion that things will remain the same unless you work this out somehow with DH. I am in the same boat. My DH will accuse me of being mean just for making SD go to bed,(at midnight), so she gets enough sleep for school in the morning, and for making her do her ONE chore,(dishes, ONE time per day- I wash all the others),. Yours sounds like a, Disney Dad", also- a Dad who feels guilty sonehow about the affects their previous breakup of their marriage had on their kids so they basically let them do,(or not do), whatever they want, and give them whatever they want if they cry. They see everything their child does as innocent and blameless. 

My suggestion to you is to try and discuss your feelings with DH. Let him know that you're not handing out chores because you,"need them to help around the house",. I don't know about you, but I don't need help doing my dishes. I am fully capable, and honestly probably do a more thorough job. I try to make my SD more CAPABLE in her future by giving her one chore at a time. Once she masters one, she moves onto another. This is how I've done it with all my kids. I don't hand out a deluge of chores- only one per day because my SD NEVER had ANY chores, so I'm teaching her like I would one of Mine,(when theyvwere waaay younger, of course),. Chores teach kids about family teamwork,(if last nights dishes don't get done we have nothing to eat off of today- in that sense SD perhaps will realize how important it is that she pitch in as part of ,"our team",. Also, letting her know that I don't send out adults who aren't capable of caring for themselves. I let Mine all know-"Someday, sooner than you realize, you will have your own house, and your own family- it's important that you get into good habits now and master these things now, before you're an adult on your own, and are responsible for ALL of the chores in your own household". "Here, you are part of a team, but someday soon you will have to do all the chores in your own house by yourself- I'm just trying to help you",.. I make sure my SD KNOWS that I don't NEED her to do  my dishes- I'm TRYING to help her learn to take care of herself, and if I DIDNT care, I'd just let her be a slob! Mine doesn't,"get it", and of course, SD is blameless, so all I can do is either stay and disengage as much as I can,(this involves going to my room alot, because otherwise I feel myself getting angry just watching DH let SD get away with everything), or I have to make a decision about whether I can stay in this state of mind. Obviously something HAS to give! Good luck sister! Know that you're not alone! Disengaging is hard, especially full time,(impossible),but if you do a little at a time, sort of weaning into it, you can do it. You have to somewhat, just to maintain your sanity!

Personally, I'd leave the parenting up to him 100%. He wanted custody, he can step up and be the parent, or send her back to BM's.

As for the dishes, as the others said, leave them for DH to do. He can take care of the ants, too.

Keeping sanity while starting disengagement is hard, but with practice it will get better - much better!

Start with the small things....like make sure DH does the dishes. "DH, you let her go to bed without doing dishes, since that is your choice, please do them now so we don't get ants in the house.  And if you choose not to do that, and we get ants YOU will be paying for 100% of the extermination cost."  Because if you have no authority to make sure your SD does things, you have no responsibility. None. I said pretty much similar things to DH over and over again regarding damage, mess, etc. left by SDs.

It gets easier.  Once you are comfortable with that first disengagement item, move on and add another, or more, at your level. I found adding them slowly was easier for me, unless there was some kind of 'event' like myself being told I was crazy for wanting kids to brush teeth 2x per day - because it took too much time and prevention of cavities was not an issue. OK, then, I'm done with that help!

Remain calm, be specific, let your DH know the ramifications HE has to deal with.