The funding of care in the UK is a bit of a nightmare.
The local authority have a duty to pay for care.
They are allowed to financially assess the person receiving the care to see whether they have the financial resources to either fund or financially contribute to their own care.
A care need and a health care need are different.
Care needs are to be met by a local authority , health needs are paid by the NHS.
The distinction between the 2 is very close but a health need can best be described as medical treatment , care can and does include medical care at a low level.
MS in itself is not a health need as despite the fact your wife cannot walk she may not require constant medical help but assistance to manage her medical condition.
The answer lies in the degree of medical help required.
The general idea is to try to allow a person to remain at home with a care package but if the degree of care required is such that this is not possible then a care home is the appropriate remedy.
I hope this helps
If there are any further points please reply and I will be happy to respond.