I’m assuming you’re flying home via Heathrow??? London has 5 airports. The cheapest way is to get an Oyster card (can get them at Heathrow and some underground stations) and take the bus. From the Clink, you will need to walk with your luggage for approximately 3 minutes and catch the N73 bus (N buses are the night buses and generally run until 5/6am) to Hyde Park Corner. Then catch the N9 bus from bus stop M at Hyde Park Corner to Heathrow. The whole journey will take you approximately 1.5 hours - but will only cost you 1.30 per bus trip. Very safe - I’ve done it numerous times when I was living in London. The Clink staff are also extremely helpful and can give you maps about how to get to the bus stop etc.
Another useful piece of advice is, check out the Transport for London website www.tfl.gov.uk
Go to the Journey Planner section. In the ‘from’ section write Kings Cross and in the ‘to’ section write Heathrow. Click the ‘more options’ section and put the date and time etc and it will come up with the various ways you can get to your location.
Hope that helps