As a Londoner who does that route a lot:
- The easiest option is to take a taxi, which have plenty of room for luggage. However, it will be quite expensive, probably around £80.
- The cheapest option (excluding buses, which would be ridiculously tedious) would be the tube, which will cost £5.50 per person each way - assuming you are paying cash (you can save money if you get an Oyster card (or use a contactless payment card), which drops the price to £3 or £5 depending on the time of day). Fortunately in your case you can take the Piccadilly line the whole way there. The tube is more cumbersome with luggage, but it's doable if you are on a budget. (check for engineering work changes)
A reasonable compromise for one person might be the Heathrow Express to Paddington, then a taxi to Kings' Cross, which is only a short journey. However, it's £34 return each, so you won't save much money over a taxi with two of you.