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Hello I am Dr RB and I am a small animal veterinarian. The area that she is defecating on may have some residual odor that is causing her to continue going there, especially if it is carpeted. I would attempt to find a cleaner that will eliminate and completely neutralize urine and feces odors and see if this helps. Given that she is an older dog with mobility issues, another solution for her would be to crate her or isolate her to a small space via baby gates while you are sleeping. This would both protect her from hurting herself/falling on the stairs as well as isolating her from her favorite pooping spot.
I hope this is helpful to you:)