I work for Just Answer as a mental health professional and also as a full time manager for the Dept of Aging. Your question immediately caught my attention when I signed on.
Capacity issues such as what you have described here with your mother can be tricky. However, I hope to clarify some things for you and offer some assistance.
In Florida, the local Area Agencies on Aging can be of immense help. This link can help you locate the one appropriate to your mother's address.
When you contact them, ask for a care manager and a level of care assessment for your mother. These care manager assessors can evaluate mom at her home (and they are used to situations like this so even if mom is very reluctant or even unwilling they often find ways to assess the risk factors involved.)
This assessment will make any services or placements into an alf (or PCH) much easier, and will determine if mom's needs should be met in an alf or in a higher level of care. The care manager can also link you to services and even offer resources for legal assistance to you so you can gain some control over this situation.
In Florida, POA is very difficult if not impossible to gain if the person is unwilling to sign the POA papers. And the person must be fully capacitated to sign a POA, which your mother may not be. Instead, a partial Guardianship may be more appropriate as this avoids the limitations of a POA and it allows person's named as guardians to make choices that are legally binding. This process is usually initiated through Family or Orphan's court and although there is some cost to it, it tends to be reasonable in expense and far more effective than a POA. (A POA is easily revoked).
Further, a care manager can determine if mom has exceeded reasonable risk to herself and is now at imminent risk. This means that Older Adult Protective Services can step in and determine if the risk is too great for her to live on her own and take the heavy burden of telling mom that fact away from you and onto them. (Better she is angry at them than family). They also can link you with physicians who make capacity determinations (and psychologists) and get the ball rolling on either placement or guardianship. If this does not happen: Yes, she needs to see a family doctor with you telling the doctor ahead of time that you are worried about mom being capacitated. Although the doctor cannot tell you anything about her care, the law allows you to tell him/her anything that would help a medical assessment of your mother.
Your brother: On a case by case basis alf's will admit two family members. However, if your brother has a chronic MH issue, sometimes the alfs will be reluctant to bring him in unless they are convinced he is stable: Or if mom has dementia...they may admit both but keep them in distinctly different parts of the facility. (Sometimes state regulations require this as well or limit admissions of MH adults, only after a more in depth assessment is completed.)
Again, here the Area Agency on Aging can help as Ombudsman (professional advocates for long term living such as alfs and nursing homes) will (if asked nicely) give you the inside track on what facilities are more open to this arrangement and which ones look nice but in fact are not. Ombudsman are wonderful resources in and after placement and I would encourage you to talk with them prior to talking with alf's of any type.
This is difficult situation for certain as mom is right on the cusp of being incapacitated by law. But with the Area Agency on Aging's help and assistance (or even a professional private care manager) this confusing system can be navigated and mom given the help (and your brother the respite) that is needed.
My best to you and your family. Steven