How do I start the conversation with my family about putting my mother in a nursing home?

You need to find a third party to speak with and for you. A social worker through a local non-profit may be able to help. They can assess the situation, and gather the siblings and let them now how things stand. Your siblings have had it made and they don't want things to change.
Also, ask the doctor for direction. Few people want to go to a nursing home, especially when a family member does everything for them, but you need to consider your marriage, your child and your own health. Please find someone outside the family to direct a meeting and get this rolling. It's time.
Good luck,
Carol

Get the family together and tell them you can no longer care for mom and that you're going to be moving out soon. Be prepared to have your ducks in a row before you do that so they can't call your bluff. Put the ball back into their court.

I have to agree with Carol. As someone who took care of her mother with 5 siblings on the sidelines never helping and outside person would have been the answer. To this day none of my siblings understand what I went through managing care for my parent’s day in and day out. Asking them for help and them actually stepping up to take over for you are two different things. Your loved one is used to YOU helping them and they will have to adjust to a new care giver as well mentally and physically, YOU as well will have some adjusting as well. As easy as you may think it will be to walk away from full time care giving I bet you may have some adjusting to do. Good luck and tread lightly, please take Carol's advice.
Blessings,
Bridget