How do I cope with putting my mom in a nursing home?

Look at this as a positive, she will be getting the 24/7 care that she needs. You and your siblings have been a stellar team. Now it is time to let go and reformulate your lives, knowing that everyone has done the very best that they could.

I wish you and yours the very best in this difficult time.

Thanks to Grandma 1954, AlvaDeer, Daughter1930. Your words of encouragement has lighten a heavy heart. I needed to hear that it will be okay. I wish you and are caregivers many blessings for this is a road that is difficult to travel.
tkx
ktmom2004

I was broken hearted when my mother had to go live in a nursing home. There was just no other choice. You must remember that you’ve done your best for your mom and you’ll continue to do so. Try to shift your focus onto what you’ll need to do to become her advocate in the nursing home. She’ll receive better care when the staff sees that she has a family that cares and is watching. But you also have to remind yourself often that your mother’s decline is the natural course of life and try to find the positive, both in memories and whatever good times may be had now. Blessings to you as you walk through this

You cope with this the same way a parent copes with a diagnosis of leukemia for their child. You make your Mom as safe and comforted as you can without giving up your own life. You understand that not everything in life is able to be fixed. You move on the very best you can and you allow yourself to mourn our human inadequacies, our limitations. You accept the tears when they come, the depression. You understand that feelings are like weather fronts; when they come in you just have to pull your collar up and endure. You understand that Saints die shot full of arrows and spending eternity trying to negotiate answering the prayers of a mourning humanity.
It is awful. It is painful. And you will want to make it your fault. It isn't your fault. It is worth crying about. It is worth mourning the loss after loss after loss that comes to us in aging the way we currently age in the world. I am so sorry for the grief, but please, in the midst of it, do not take on self judgement, nor the judgement of those who have not a clue in the world to what you are going through.
Guilt is for those who murder, who steal, who torture and who do so with malice aforethought. Guilt is not for good and decent people with limitations. Hugs out to you. There is no way to survive the buffeting of weather-front-feelings but to get through them. And then go on with a decent life. THAT is what any good Mom would want for you.

Look at this with the idea that you are trying to get her the best care possible.
Admitting that you can not "do it all" or handle the emotions and all the stuff that goes with it is not a "failure".
When you are a caregiver the dynamics change and the relationship changes. Having someone care for her you can become a daughter and not a caregiver. You can manage her care, you can be an advocate for her.
You can go hold her hand, take her for a walk or maybe out to lunch and enjoy the time you spend with her and not worry about all the other stuff.