I'll try and keep this as short as possible.
So what exactly does your husband do at home all day?
What kind of conversations have you had with him about housework, job hunting, etc.? How do they go?
(edited because I misread it)
He says that our 15 month old daughter is so hard to handle and requires his constant attention, that he doesn't have time to do anything.
I do not have children, so I may not have the best point of view.
I am sure there are stay-at-home moms that watch one toddler yet can still keep the house somewhat orderly and have clean clothes. To other it can be overwhelming. I have seen others that feel lucky to even get in a shower on the other hand...
It just seems so unbalanced for you to work 2 jobs as many hours/week as you do and then come home to cook, clean, laundry, bills everything. I mean what if you gave him the same line, "I'm so overwhelmed with work I just cannot make any more dinners." It wouldn't work I bet. And that he doesn't want to remedy it and let it go as long as it will is not a good outlook.
What about having a serious family meeting? Start off by saying that he's a great dad. Give some praise of some sort. Make a list of all the things that you do everyday/week for the household. Show him how imbalanced it is. Ask him to help you figure out a better way to make this more doable because at your current rate you are fried. Come up with a list of expectations for each. Make sure the meeting ends with duties that he has to do and agrees to do. Say it's necessary to keep this family going.
hHave you tried that approach? It's different than coming home and saying, "OMG, why are the clothes still sitting in the hamper???"
You could also mention in your meeting that you find it difficult to be intimate with him because a) you are so busy with house chores at night that there is no energy left in you, and b) he's not fresh daily.
So now - if he were to be super Mr. Mom, do all the chores, have a spotless home, always freshly showered/shaven, greet you with a smile and a happy baby - do you think life would be dandy? Would there be more issues? Would you be attracted to him again? That would be an interesting answer.
I'm thinking there is resentment that he's spending time with the baby while you have to do the traditional provider role.
Is he saying if you were more intimate he might help out more? Was that in the equation?
Well, I don't know if it's too late for answering, but here's the opinion of a housewife with a strong-willed toddler. I was a professional who is a housewife because of circumstances and not by choice, meaning that I don't feel happy staying at home and doing chores, I'm not good with kids and I think it's one of the most difficult jobs I ever had (and I've been working since I was a teenager).
My 2 year old is very strong willed and when he was 15 months old, I honestly couldn't do much either. I had to constantly watch him to prevent him from harming himself, he would refuse to do many things, sometimes it took 15 minutes to change a diaper, because he was running away or sitting down, or I couldn't get him in his sit, or he would throw food on the floor. He would wake up once or twice during the nap, so I couldn't cook (well, sometimes, depending on the dish). Whenever my partner stayed with him, he would say that he was easy, because he would just play with him for a short period of time.
But I could manage cooking (I would do it at night when he slept, because he would always want to be in the kitchen glued to me and I felt it wasn't safe), washing dishes, most of the grocery shopping (but at some point, I had to have a babysitter to be able to go), laundry and some cleaning (the kitchen and whatever food my son would throw on the floor). But the house wasn't very ordered, because I'm not good at ordering things, and whenever I tried, my son would remove clothing from drawers or throw other things and honestly I just felt overwhelmed. Then my partner would come home from work and ask what is it that I did all day and criticize me. Didn't help the relationship....BTW, I didn't take showers everyday either :).
I agree that it's not reasonable for you to be responsible for all the chores. There's no excuse for not being able to do the laundry for example. Your daughter can even "help". If you take care of your daughter when you come home, he should be able to do some chores, that's when me and my partner did. And when you're home, you should share.
Your husband could be depressed about his situation as well. Maybe he doesn't know how to organize himself at home. I know that if my partner suddenly had to be a stay at home dad, he wouldn't do 50% of what I do ( I was hospitalized for 5 days, he had a person who was coming to clean and cook and he was still overwhelmed with our son without me).
Honestly, if he's responsible for taking care of your daughter and will have to do chores, I don't think he'll have much time for looking for a job. He might be able to send some resumes at night, but it's not a very efficient way of looking for a job. Either your daughter has to go to the daycare or you need someone else to help out with watching her, while he's looking for a job.
I wouldn't criticize him for your daughter being too attached to him, it's normal at this age. My son would sometimes refuse to be with anyone else than me, then he would have a period when he wanted my partner, then he would want to be only with me again. That's just the way they are.
Try not to blame your partner and put yourself in his shoes. I'm telling you that it could be very hard with a child of this age. Try to talk to him and find a compromise on chores and try to find a solution so that he has time to look for a job (if that what you both want).