How to finish PhD and get through the academic job market as a new mother and primary caregiver?
Thanks for asking this important question! I will address three aspects of surviving your next year as a job market candidate and primary caregiver: changes in research style, finances, and emotional balance/accepting help with childcare.

Before launching into it though, I do want to emphasize that things get easier. Eventually, baby learns to sleep.

Research/work style: As someone who is juggling caregiving and research, you need to be find a way to use small chunks of time. It will be much harder to get long uninterrupted stretches of time, so train yourself to do without that. Break your work up into as small of chunks as possible, make a list of those chunks, and try to get to them one at a time. This may be more difficult if you are in a lab science and the only place you can work is from the lab. However, you may be able to write at home. To write, I would suggest starting by writing an outline. Then write the topic sentence of each paragraph. Each time you have 10 minutes to concentrate, write one of the paragraphs. 

Finances: The best piece of advice I got as a new mother/academic was: throw money at problems. This is not the time to be frugal or save for a rainy day. This is the rainy day. Your career path will look very different depending on whether you finish this PhD, so you should throw resources into finishing. If you can borrow money from family, do it. If you can call in favors, do that too. You should focus your spending on things that buy either sleep or time - i.e. good childcare. 

Childcare: If you plan on staying in academia, I think it is best for you to learn to embrace the positives of childcare. There are great childcare options out there, and I found that some childcare providers were just as good with my baby as I was (some probably better because they actually had more experience!). Take the time and use whatever money you have to find someone good, either for in home and daycare. This is where you want to spend because it is valuable to have peace of mind. Honestly, as a mom who has seen development up to age 2, I would say that my child cares if I am around a lot more now (at age 2) than they did at the beginning.

Furthermore, when you are home, embrace the "electronic babysitter" (the TV). Sure, it feels bad, but at least some quality research shows (http://www.slate.com/articles/business/the_dismal_science/2006/02/the_benefits_of_bozo.html) that TV doesn't harm kids. Now most programs really don't help them either, but you can get a couple of 30 minute work sessions a day using the TV without guilt. My daughter loved Fantasia (lots of fun colors and classical music!) as a baby. Now she loves Daniel Tiger, and that program has been shown to increase preschool readiness (http://www.pbs.org/parents/expert-tips-advice/2016/07/daniel-tiger-helps-teach-social-skills-preschoolers/). 

Finally, if you absolutely can't use a babysitter, embrace the two-parent household you have. As soon as your partner gets home, you leave for lab. At least one day per weekend, you are in lab. Reserve maybe one-half weekend day for family time. Yes, it will suck. You won't get time with your partner while you are in this stage. But you will get more work done, and that may lower your stress. 

P.S. The best thing I ever did to get my daughter and me sleep was moving her to another room. She was sleeping in my room and we woke each other up all night. (She snores, my husband snores.) She slept for twice as long at a time once she was in the other room.