Allsorts Do you know if your neighbour is eligible for funding from the council? If so, she can apply for Direct Payments - you then get to choice the care provider as opposed to the council choosing on your behalf (which I assume is that she meant by "NHS one"). I know the eligibility criteria for people who self fund their care is assets and savings over (roughly) £23,250, anything below that is part funded by the council on a sliding scale until care services are fully funded by the council (below £14,250).
Worth checking out if she is eligible (contact the adult social services in your area and as part of their assessment they will also do a financial assessment). In terms of private (self-funded) care, your best bet on finding a good agency is word of mouth recommendations and online reviews - Homecare.co.uk is the best for this (they verify all their reviews before publishing them, and they are coming from the person receiving their care or their families, unlike google reviews that can be faked).
Cost of care varies based on where you are located, however you are looking at starting from £23 p/h for a care provider. You can hire your own carers privately, but this will require more vetting from yourself as this is an unregulated market (unlike care providers that are regulated by the Care Quality Commission, aka CQC).
Remember, if you are unhappy with any aspect of your care from a care provider, you can raise this with CQC if the provider is not taking the appropriate actions.