3-4 hours from Heathrow gives you most of the country, assuming you mean by car not on foot, and that you mean to go there and stay, rather than spending your time in transit. 5-6 hours would give you all of England & Wales, and some of Scotland as a bonus (spell-check just stopped me saying Sotland, hmmm).

Three nights almost anywhere is likely to be good, if you go there then explore a bit, the inside of a car varies only in temperature and humidity, while what passes the window is available on TV.

I can't help with the football thing, at some 50-odd years (some very odd) I've yet to attend a match.

Go north or west, for choice, IMO. One day I will have a look at Kent, it sounds nice, but above all don't spend your time travelling.