The least expensive method is the Piccadilly line on the tube. At the tube station at Heathrow, you can purchase a ticket to KC. Or, if you plan to return from York and visit London, you can purchase an Oyster card. The Oyster card is only good for areas around London but the agent will add on any extra cost to get you to KC. They did for us. About an hour to KC.
If you can get to KC with some time to spare before your train departs, the fabulous British Library is immediately on the other side of the St. Pancras train station. Just go out to Euston, turn right and walk a long block past St. Pancras. It's one of the few things in London that's free to enter. Great book rooms. The Magna Carta is in there also.