The other issue with New York is that hotel prices can be ferocious. BA holidays can help a bit with that (and I hear that hostels are often surprisingly good).
Personally with limited time I'd look shorter haul, there are some interesting places, North Africa for example, Malta, Eastern Europe, the Baltics, and they're well within range without spending half your holiday in the air. I'd say Beirut except your dates don't work for that unfortunately.
There are amazing deals to the US if you bide your time. I'm going in early June on the back of the £100 for 5000 avios offer, and with that and the price promise and the net avios, the fare works out at about £85, with two nights in the Conrad on Hilton points. If you do it on a whim you're not likely to hit the confluences you need to get this sort of price level.