Bookmarks- I had this exact situation with my mother a few months ago. It's so overwhelming, especially if you're on your own with it.
The hospital really must not discharge her if it's unsafe to do so, especially as she lacks capacity. Use the words 'unsafe discharge' and 'fall risk' in your discussion with the hospital, to buy yourself a short time to consider the options, bearing in mind there will be pressure to move her to free the bed. They can't discharge her without your agreement if she lacks capacity but pressure will be brought to bear.
Has her infection completely cleared? That often causes confusion and falls. How old is she?
You should also be put in touch with an adult care social worker, who usually in my trust area at least, acts as liaison between the various agencies.
Your mother should have had a proper assessment from a physio to try and get her mobilising again. She should also be referred to the falls team. Check with the hospital if any of this has been done. I would consider a preliminary discussion with PALS too as she fell whilst in their care having been informed she was a fall risk.
The problem with dementia patients is that bed rails are not recommended as they can cause injuries when the patient tries to climb over etc.
Wherever she goes after this I would look into having her provided with a raise and lower type bed that can be lowered close to the ground, then she can have a fall mattress to the side and foam blocks to support her to minimise the risk of injury.
In my area this equipment was ordered for installation in her care home room via a request from the hospital to the district nurse team as part of the discharge process. I am not sure if this equipment is available on a ward - I doubt it- but it might be worth asking.
Assuming there are no further undiscovered injuries as mentioned above, and if the physio supports it, they might suggest a short stay in a re enablement unit ( the trendy word for rehab).
If it's found she does need one to one care. Is she self funding? If so, look at the funding options. NHS continuing care might be a possibility, but won't I think cover all the cost. It's sometimes possible for funded one to one to be provided in a care home to fund the extra carers needed, but that depends on the home.
Have another discussion with the care home manager to understand their exact position on all this and why they feel they can't manage her. This may well be an entirely sensible safeguarding decision on their part but it's worth checking again what their thinking is, ( I don't quite understand the wheelchair point - a lot of oldies are in chairs but not completely compromised) and if there are any options you aren't aware of.
Have a look around at local nursing home options anyway. If she recovers and goes back to the care home it is still useful knowledge to have for the future.
The information above is learned from my own long and bitter experience with two sequentially ailing parents over a couple of decades . The processes ( which may of course vary somewhat from local authority to local authority) did not always work as they should have done. But I only knew that in retrospect.
Knowledge is power. Good luck, and look after yourself 
