Both are reputable universities. Durham has an excellent domestic reputation for its English department and in recent years has been a league table-topper within domestic ratings.
However, in terms of international reputation and 'brand' prestige then King's College fares a lot better. King's is among the most prestigious group of UK universities for its name and its age. However, King's does get a fair number of complaints about teaching quality and the quality of student life (its central London location takes away from the 'student town' atmosphere that a lot of UK universities exhibit and are popular for). Of course, it produces excellent, world-class research and its lecturers are distinguished individuals, but many also feel that its prestigious name inflates its reputation which many believe is more than its teaching quality may deserve in comparison to other prestigious UK universities.
In contrast, Durham is touted for its teaching quality and student life and has earned itself a positive domestic reputation. It's not bad internationally, just not as distinguished to American, African, Continental European, Australian, Asian employers as say an old London university.
I hope my reply isn't too late, I'm not sure what international admissions processes are like but domestic applications have pretty much closed by now in the UK apart from Clearing.
As I've already mentioned, King's' Central London takes away from a student atmosphere. However, it is very conveniently located, it's not difficult to catch a bus and reach Britain's largest coach, bus and train stations from any of King's College's main campuses, and I think it would take no more than an hour's ride on coach to either Heathrow or Gatwick airport, Britain's largest airports.
I admit living expenses in London are in fact much higher than in the rest of the UK. Public transport can be double the price of average UK public transport prices, although food, drink and clothing from large chain retailers are usually kept at standard prices in London as they are in the rest of the UK, so it's mainly transport and accommodation that's expensive. Try saving by registering for a student's Oyster photocard, you get I think a 30% discount on major public transport links.
I've never been to Durham but transport and living costs should be much more affordable there. You also get a lot more of Britain's student atmosphere, HOWEVER as an American you may feel more comfortable in London as Durham can, I admit, feel quite isolated and foreign and you may not feel as comfortable as you would in London where you can conveniently hop on a plane and go home.
Newcastle isn't too bad in terms of culture. You'll get plenty of access to theatres and cinema and all sorts of entertainment. From what I know, the area of Newcastle does have an ancient history (longer than anywhere in America, obviously) so you can always visit libraries and museums. However, as culture goes you can't beat London. I already mentioned that London has accessibility and resources as an advantage, and it does. As an English student you may be interested in watching performances at Shakespeare's Globe, and numerous national theatres. There's the British Museum is you're interested in European heritage. Release your consumerist side in London's wide range of commercial areas. You can go bohemian chic in the famed Camden Market, or high street savvy in the equally reputed Oxford Street. Newcastle isn't far off in this respect though and you may find it less crowded and confused there.
Depends on what you like as a person I guess, but I personally wouldn't dream of going so far up North. Take regional accents into account as well. As an American it would probably be difficult for you to comprehend the Northern dialects and their thick Scottish-influenced accent in Durham which can pose a great range of problems. London accents will probably be much more intelligible to you.