I stayed at Holland House youth hostel, it's an HI hostel. Nothing fancy, but it was a pretty easy bike ride out to Heathrow to catch my flight the next day. They let you store your bike in their very secure luggage storeroom, which is nice. It's probably the closest hostel to Heathrow and the setting is pretty quiet for London (fronts on Holland Park). I'm assuming you're en route to PBP, will your bike be boxed or will you be riding it around London?