She could make a homeless application to Birmingham council.
Even though she has a roof over her head, she can be considered homeless on the grounds that her current property is unreasonable to occupy because of her health conditions.
She wont automatically qualify for emergency housing as she is not automatically priority need due to having no dependent children that need housing with her.
The council (under homelessness) can look at her vulnerabilities and if they deem her "more vulnerable than the next ordinary person on th e street" then she may qualify for emergency housing (hostel etc) whilst the council investigate the rest of her application.
If she doesn't qualify for emergency housing - the council will still have a duty to "relieve her homelessness" This duty lasts for 56 days and they may help with a deposit on a privately rented property during this time. Or they could offer a private rent as long as it is available for at 6 months.
If she is owed the main housing duty it is VERY unlikely that she will be offered a council property in Birmingham as there isn't any. ( I deal with this council regularly) She will more than likely be offered a room in a privately rented property.
If she is currently in a council property where she is, I would strongly recommend that she does not give this up lightly as she is unlikely to achieve the that security of tenure again(especially B'ham)She would be better waiting for a mutual exchange although I appreciate this can take time.
I am surprised that she is allowed to bid on the Birmingham waiting list as Birmingham allocations policy states that to join the waiting list the applicant must have lived in the borough for at least the 12 months.
It's not going to be easy op -sorry to say.