There is no easy answer to this, and as much as I despise Johnson, he has to be given some credit for trying to find a solution. A solution that will no doubt have some section of society up in arms.
The ideal of the Welfare state was that it was a system that worked from cradle to grave, which was a noble ideal that has been eroded since the 1970s by successive Governments of all hues. There is so much interlinked with this it is a minefield. The destruction of the Social housing base which encouraged people to buy their own homes led to a housing price boom and many did benefit from that. Life became about personal responsibility rather than collective responsibility. Older people with their own homes had more equity as a result, which meant in effect that pensions could be lower. The UK has one of the lowest pension rates in Europe, so older people can be asset rich and cash poor. When we had Social housing people would be asset poor and cash richer. The obvious way to raise money for Social Care therefore is taxing asset wealth. People who have become asset rich, have done so because society has provided the conditions for them to become asset rich so I would argue it is only fair that those who are asset rich bare the greater burden.
In my idealistic world Social Care would be a branch of the NHS and free at the point of use, that though would mean nationalising Care homes, setting a minimum standard of care and removing the profit motive from Social Care. If pensions were payed at a rate that was greater they could be used as basically bed and board for a Care Home. That way those who have saved will be able to leave an inheritance, those who were unable to save will still receive Social Care. This would paid for by increasing income tax, the state spend is currently one of the lowest in the Western world and by increasing the spend by a few % of GDP raised by tax rises this could be achieved. I am sure people would pay an extra couple of pence income tax if they knew that any future Social Care would be provided free at the point of use. Although I am not so daft to think that many would be against this because RW think tanks like the TPA would be aghast at the thought of people actually contributing to society due to there infatuation with individual responsibility. What good is individual responsibility though if you are struck down with dementia and can no longer be responsible, if you had paid those extra couple of pennies income tax you would not have to worry and you would not be a burden on anyone, you would be entitled to Social Care until the grave.