How do I go about getting my brother into a rehab facility?

How did he get in this shape? I am older than he is. Perhaps he has depression. ??

A doctor would need prescribe. If he is up satisfied with his doctor he can / should change. A doctor will likely only prescribe if medically needed.

sometimes there is no help till they end up in hospital

A primary care orthopaedic physician who treated him for the falls could prescribe PT.

Is your brother weak because of specific medical conditions or because he's not able to prepare healthy meals? Or is he diabetic? Either way, Meals on Wheels would help if he qualifies. You could also get him some Boost or Ensure to drink. Opening a can is easier than preparing a meal.

I'm not sure a social worker could prescribe PT, but one might be helpful for his general overall wellbeing. Ask his PC doctor if he has one.

PS. If you think it's a matter of nutrition and physical exercise, you could consider hiring part time help a few hrs a week or arranging for MOWs.

You have to find a doctor willing to work with you and can examine him and order PT or rehab. It doesn't mean that he will be admitted to a rehab facility if he won't get better from rehab. Doc may order Physical therapy and have a PT visit at home or suggest brother go to PT outpatient facility.

Usually a doctors office (geriatric care, internist, etc) sometimes has a social worker on staff.

Otherwise, visit your local senior center, etc and see if they can recommend someone to evaluate brothers home and living arrangement.