I would second the view that, if money is not the issue, do it overground by car. Whilst there's not much to see, at least you are seeing something. And it's far more relaxing, whether or not you get stuck in traffic.
Otherwise, Heathrow Express plus changes will take just as long or perhaps longer than the Tube, be more stressful and will cost many times as much, so should be avoided.
If it were me, I'd do the Tube and spend the money on a treat, but then I live in London and I'm on the Tube every day anyway so it's no big deal. You will almost certainly be using the Tube as you get around London, so if you take a car, you won't be missing out on a London experience.