I suggest you consider flying between Heathrow and Edinburgh. A short inexpensive flight where the train is 5-6 hours, and as was stated above, you have to go into the London city center to catch the subway to Kings Cross and pick up your train there. If you're already at Heathrow.....
If you do take the train, consider spending one night in York, an old historic town with the incredible Yorkminster, a church that looks like Westminster Abby in London. Walk the medieval wall and go into Betty's for coffee and pastry...but take plenty of money. You can get a reasonable one-night B&B in walking distance to the city center.
If you fly, take the Airlink 100 bus into Edinburgh. There are several Airlink busses, so make sure you get the 100. It cost less than £6 one way and the final stop is a short walk on Princes Street to Waverly. I was just there for three days in July, and because we had golf travel bags and after visiting Edinburgh were renting a car there to go to go to St. Andrews, we stayed in a hotel by the airport, and used the Airlink 100 each day. Round trip was £8. Another option from the airport is the Tram, but both have multiple stops and take about the same time. I recommend the bus because you get to see the outskirts of the city on the way in.