In all of these discussions I say the same thing, so sorry if you've heard this before, but IMO nobody should be paying for care, as we should all pay more tax. Every adult should pay, and get back when they need it. Those who choose not to pay should still be provided for, but only if they need it, and then they should be means-tested on their household income, or whatever it was that paid for them not to have to work (a trust fund or other financial provision).
I am not talking about those who don't work because they are caring for a disabled child, or a sick partner, or those who are themselves ill, disabled or otherwise unable to work; but those who choose not to do so, either because they have a partner who 'supports' them (so one partner pays one lot of tax for two adults and their children), or because they are rich enough not to need a salary to get by. Anyone unable to work should be credited for as long as necessary, but those who benefit from an education, a health service and all the other things paid for by tax and NI, but opt not to contribute should, IMO, have also opted out of the benefits, such as a pension and social care, unless the source of their support is no longer available.
Under that system, it wouldn't matter whether someone has a house or not, whether it is more or less valuable or how much they have saved. Everyone would get the care they need, and a decent pension in older age.