CHC fully funds care, but I would presume they would have some say in where the person is placed.
The apprx £500 you quote is the Social Services limit on payments to Care Homes- completely separate.
Nursing Care is currently £159 per week if classes a needing nursing care but not qualifying for CHC.
As lemonjuice says the care is fully funded and there are no limits, however, the NHS has to find somewhere that can meet all her/his needs as effectively, the NHS are in effect contracting out the care to the private sector to free up an NHS bed and as local authorities (in most of the country) no longer provide care like this it goes to private sector care homes.
There should be some consideration given as to being local for family and friends to visit and you should be able to pick somewhere you like the look of or keeping her where she already is but the reality may be different although as I understand your mum is already in care that might not as big an issue as someone entering the system.
K