There are lots of hostels in the neighborhood where you can stay for as little as 16 to 18 GBP (under 20 Euro) per night for a bed in a dorm room.
As long as there is no Tube strike, you could also think about staying in a hostel in any part of the city where you can connect easily to the Piccadilly line that goes to Heathrow. The Tube takes a bit longer, but it is cheaper than Heathrow Connect if money is a concern.
You have to pay to leave your bag with the left luggage office at Kings Cross, and you have to pay for the showers and toilets there. Why not just pay a little more for the hostel where the bathroom/shower and luggage storage are included and you'll have a real bed with no chance of the police kicking you out?