Dear Irene,
You overcome the guilt by looking at the situation logically and not letting emotions guide your thinking. When my mother got Dementia, it was my greatest wish to keep her at home with me and not 'put her in a home.' I thought I was doing it for her, because she would be happier and not as confused and not under so much stress if she stayed in her familiar apartment where I lived with her for 5 years.
Now I wish we had allowed her to be in a good nursing home at least 1 year and maybe even 2 years earler, because it turned out that living at home was far more stressful for her than livng in a nursing home. I was still working at that time, and Mom was alone much of the day. I didn't know that she was wandering around the neighborhood (in Queens, NY) looking for her house, was becoming unpopular at the senior center where she went because of her bizarre behavor, was being taken advantage of by a bank where they tried to sell a 95 year old woman a long term annuity, was unable to decide what to wear every day and so was going out in the winter with summer clothes on, and a million more examples. When we got a home health aide to stay with her while I was at work, she felt like a prisoner and didn't accept at all the idea of having 'a stranger in the house'. I had no idea she was under so much stress and pressure, believing she was glad to have choices, glad to be taken out to the senior center to be among other people, etc.
When she finally did land in a NH after a fall and broken hip, she became so much happier! Someone came in and washed and dressed her every morning, gave her food and helped her eat it (I had still been under the delusion that preparing her own food was good for her) and above all she was safe. She could also relax and stop trying to convince us all that there was nothing wrong with her. It took a few months to adjust, but now she feels entirely at home there. I must add that the nurses and aides are exceptionally loving and caring.
So - don't feel that you did anything against your Mom's best interest; on the contrary, allowing her to be a resident in a NH was the best thing you could do for her - she is safe, warm, properly dressed, given her medicines without having to remember to take them, examined by a staff doctor once a week and any infections nipped in the bud, has the company of other people if she wants it, and is free of the stress of having to make choices.
Good luck !
Martha