First of all, keep in mind that I am unable to diagnose you because I am unable to examine you, this forum is for educational purposes.
The symptoms and story that you provide are consistent with a progessive cognitive decline, which can be due to a number of different causes. The most concerning of course is a progressive dementia (of which there are several types), that are all irreversible (treatments are unavailible or do not stop progression of the disease). So the first job of physcians is to evaluate for treatable causes of cognitive change, which include depression, vitamin deficency states, metabolic illnesses such as infections, and vascular disorders (such as strokes, etc.). This is done by checking blood work/urine for the vitamins (B12, B6, Vitamin E), infections, copper metabolism for Wilson's disease, etc. and an MRI of the brain to evaluate for a structural cause. Depression is more tricky, there is no blood test for depression and the diagnosis is usually made based on clinical judgement. In cases of cognitive change, depression is often evaluated by a trial of antidepressant medication to see if things improve, inaddition to Neuropsych testing which can sometimes identify depression. The second job is to quantify the deficit with clinical testing in the neurologist office (often not very sensitive) and formal neuro-psych testing. Testing should be done more that once to get an impression of the changes over time. Often high functioning individuals will score 'normal' on the tests, but they have actually lost a good deal of function to get to that 'normal' reading. Repeating the test can show if there is a progressive decline in function, or if there is a response to anti-depressants/vitamins, etc. Third, if dementia continues to be the most likely diagnosis then refining the diagnosis is next. The most common is Alzheimer type dementia, but others including fronto-temporal dementia, Creutzfeld Jacob Disease (CJD), lewy body dementia etc can be tested for. A PET scan can show regions of the brain that are not metabolizing as well and is often used to differentiate alzhiemers from fronto-temporal dementia. CJD is less likely in your case (given your 3 year progressive course-it is usually much faster), but is often tested for with an EEG (electroencephalogram-or brain wave test), and lumbar puncture for 14-3-3 protein and neuron specific enolase. Treatments depend on the diagnostic evaluation described above, but if dementia of the alzheimer type is found then medication like aricept and namenda are thought to slow th progression of the disease, but do not reverse it. Given the level of dysfunction that you describe, with falls, personality changes etc, it would be wise to have a home safety evaluation, and she may need to be re-located to live with family, etc. to prevent her from harming herself. In summary, I would suggest repeating a formal neuro-psych evaluation, screening blood work (if not already done) for B12, B6, Vitamin E, Copper, ceruloplasmin) a PET scan (FDG-Glucose PET), EEG and lumbar puncture. I would recommend continuing the anti-depressant and would suggest a formal evaluation by a psychiatrist (to optimize her therapy for possible depression).
I hope this has been helpful.
If it were my mom and would never give up (though stressful and frstrating) going to doctors. Marked personality changes can come from many things, inlcluding mini strokes, Has she been chcked for this, thyroid , immune, lymes. Anything can change the central nervous systems can affect a person. I am not sure if this fits but there is aslo a disease called huntington disease. I wish you well and much luck in taking care of your mom.
Sorry to hear about your mother. You should have copies of the MRIs examined by another Neurologist. Sometimes there's evidence of frontal lobe atrophy (Pick's disease) which may be subtle, also something like Normal Pressure Hydrocephalus may be missed if not thought of. I would suggest asking a Neurologist to perform a PET scan to see if your mom has some evidence of dysfunction, possibly suggestive of early Alzheimer's disease... I don't mean to throw out diagnoses, but more searching should be done -- it certainly sounds unusual, although, not unheard of for depression to manifest itself.