Social Security, Nursing Home custodial status, when does puny needs allowance for mother start?

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Hello,

After caring for my 89-year-old mother for seven years, who has Alzheimer Disease, which I managed with a pretty controlled environment, including daily in-person checks at her apartment, for the last seven years, I feel like she was stolen from me, her life, her cat. She was admitted after falling due to running to the toilet for three weeks over a UTI. I expected she would get her usual three weeks of rehab and be sent home as with all the other times. (Last admission was 17 months prior, for the same reason. She was not a overly reliant on the ER or skilled nursing.) I never saw it coming!  And I do not feel the nursing home did enough to rehab her, but short of hanging around from 8 to 4 and seeing what was being done, I would not have been able to intervene. In short order, by doing what was convenient for the home, putting her in a wheelchair, they converted someone who was upright with a rollator to someone who no longer needs to use her limbs at all, except to eat and toilet.

She converted from Skilled Nursing to Custodial in July. Since Social Security is paid for having lived through the entire prior month, what was paid on July 3 was for June. However, how is a split month treated? All I could find online was that the benefits will continue to be paid to her checking account and then I save out the needs allowance amount and pay the home the rest. (LTC Medicaid application in process but there was a break because her renewal month for community Medicaid in a two-year cycle was this past June, and her Agency on Aging worker blew off my three calls related to the application PRIOR to the fall on Memorial Day evening.) So for August 3, the benefits for living through the month of July were paid to her checking account, less her Part B premium because her QMB/Medicaid case closed. But it is also the month she converted from having Medicare pay for skilled to nursing to LTC Medicaid-pending. So when the home sends me a bill at some point, I would like to know if she was put on the needs allowance for July or whether that will start in August.

Thanks!