Some local authorities provide free or discounted gym membership.
One thing I would suggest is taking a look at The Chronically Sick and Disabled Persons Act 1970.
Under that act, local authorities have an obligation to provide a range of services including practical assistance, home adaptations and the provision of or help with procuring access to recreational activities (such as outings, TV services or educational services, and the provision of meals at home).
This place has a good article about that legislation, which I believe can be used for gym membership:
And here is a link to the legislation:
If you are also your daughter’s family carer, you are entitled, under The Care Act 2014, to a carers assessment, the local authority may provide leisure or gym membership for free, but is at their discretion.
Care Act Section 10:
Hope some of that is of use to you