Well this is intriguing, I have a few points to make, helpful or not I am not sure...
1 My sympathies to you Debjud I have been and still am sort of in a similar position.
2 My philosophy has been that they should learn to manage things for themselves, but if they didn't and the school complained then I would step in and make sure they did the work. This worked fine with DD1 who is now at a good University. We had one phone call home expressing concern that she hadn't done a past paper, but as she had spent the weekend doing one I assumed she had set things straight (and she had). DD2 has been a different story, serious concerns over coursework at GCSE and inadequate or non existent homework in 6th form.
3 My advice would be if the school are complaining then don't worry about your relationship insist she does the homework. Likewise you can't leave it up to her to decide what should be done, although if the school haven't complained you could tell her "well I think you should do it but it is up to you, if the school complains then I will insist". Don't be hoodwinked over coursework, check with the school if you are unsure about what needs to be done, I found it was the last improving exercises that got neglected rather than the bulk of the exercise.
4 I am amazed that so many of you know what homework has been set! Do your children have two week timetables? I find all they put in organisers is "next lesson" which is meaningless to me especially when they have two teachers for some subjects. Or do your DCs actually tell you what is due in? Seeker you and your DD sound incredible, I can't believe that she is willing to do it!
5 Evil Twins your point is interesting, my DD's school seem to be of the opinion that if she hasn't learnt the skills by now, she never will, but you are saying (you have said elsewhere as well) that it is quite commmon for them not to have the skills.
It is not easy is it Debjud when they just think they don't need help?