You can get a flight from Manchester dates-Depart: Friday-July25, Return: Monday-July 28 for £164 (Per passenger including taxes) check Icelandair for tickets.
Is best to fly from Manchester it take only about 2h30 hours on direct trains, saves you going all the way to London plus half the cost. Check
www.nationalrail.co.uk for tickets, the tickets are about £65 with a open return from Newcastle to Manchester.
It will cost you alot more then that to go all the way to London I checked some and it was was about £120 newcastle to gatwick depending on the times, I don`t think is worth it!
So my advice is to try Manchester, that works out about £229 train and flight.
Booking as early as you can gives you the best choice of fares, make sure you book your hotel before you go as rooms are so expensive there, well everything is expensive there! Hope this helps, have a great time!