I'm visiting London alone, and I'm looking for a cool place to stay where I'll be walking distance to everything, and where like-minded travelers or locals will be hanging out, looking for a pint and conversation.
I know a hostel will provide me exactly this, but I can afford a hotel, so I'm basically looking for the best one for a young traveler looking to meet some people.
Nowhere is within walking distance of everything in London. Try to get accommodation near an Underground station, where there will be buses that might be good too (but their speed is limited by heavy London traffic). Also consider a few day trips outside of London by coach or train. Consider an open top tour bus in London to get you orientated as to where everything is.
I checked lowcostholidays
and using the Hotel Only and London - Central London options there are a fair number of places starting from just £100 for a week in April for one person. There are many other companies offering similar deals so shop around.
Be careful of pick pockets in crowds, even on public transport.
Nowhere is within walking distance of everything in London.
That rather depends on what you want to see in London and how far you're prepared to walk. I agree it would be sensible to be near a tube station but if you stay in Westminster or the City you're going to be near one wherever you stay. Personally I think trying to walk as much as possible is a great idea, you'll see much more of the city that way, just arm yourself with an A-Z and a good guide book.
Hotels in Central London tend to be poor value and the cheaper ones tend to be in clusters away from the centre of the city - you don't say what your budget is but you might be better off getting a room in a decent hostel (the new YHA Central London for example), this would give you the social experience you're looking for too.