Honestly, the PicadillyPiccadilly Line is what I'd do. I'm not sure how you feel you'll take two trains to get there tho - from Heathrow, depending on the terminal it's direct all the way to Kings Cross - 23-25 stops. (Takes about an hour). I've done this very trip myself.
In terms of luggage, that line is the arrivals line - people expect the luggage, and unless you're arriving at rush hour, you're going to have the train mostly empty until you reach the downtown. Getting on it WILL be empty, so you'll have time to adjust your bags and move them to the side to not bother others at all.
The station itself (Kings Cross) has had a lot of recent refurbishments, so there are more escalators and elevators than there used to be be. From there you can either walk, or grab a cab at that point, depending on where your accommodation is.
The tube is easy, fast (especially in traffic it can be faster) and cheap. Definitely the preferred option.