No doubt, you made this promise in all sincerity, and it's important that we never make any promise lightly or turn away from it casually.
But God understands your situation and knows what is best for your mother. Furthermore, He wants you to do what's best for her -- even if it means turning away from what you once told her you'd do. After all, why did you make this promise to her? You made it because you love her, and what you were really promising her was that you'd do everything you could to take care of her if she became incapacitated.
But her situation has changed, and now the best way to fulfill your promise may be to place her in a facility that can give her the care she needs. Ask God to guide you as you make this decision, and if the best thing is to move her to a place where she can get better care, don't hesitate to take that step. Wouldn't you want your children to do the same for you? Remember Jesus' words: "In everything, do to others what you would have them do to you" (Matthew 7:12).
One reason people resist going into an assisted living or nursing facility, I've found, is because they fear being abandoned. Don't let this happen with your mother, but visit her often and let her know you love her deeply, even as her memory fades.