I agree with your principle that very wealthy people should not be entitled to government hand outs, but I do not agree with you when you say "paid for by the government" (ie you and me). If she has amassed a £40m fortune then she would have paid tens of millions in tax over the past 20 years. This is far higher than "you and me".
The money she has paid in tax would of helped pay for "yours and mine" hospitals, schools and a lot of other public services. I have always found it a bit ungrateful when these people who pay vast sums of money in tax and then get slatted when they dare take anything back out of the pot. I also find it equally strange that people who earn low wages and contribute very little in the way of tax and then take a lot out of the pot, are held up as "hard working" pillars of the community.
This. If we agree that we enjoy living in a society where we look after the less fortunate, irrespective of wealth or status, then this argument is essentially based around jealousy of those with more money than us.
The alternative is that we go the US way - pay lower taxes and have a poorly-funded safety-net for those who can't afford to do it themselves.
Personally, I much prefer to have a system where essential care is open to all, and would hope that if my children needed special services getting them to and back from school, that they would be offered exactly the same support as all other children with the same problems.