I've actually found the UK a pretty reasonable destination in recent years, especially outside of London.(My budget was way under yours for a month long trip to Europe last June.)A lot of museums are free. Transportation costs within the country can be reasonable too.Not only do they have advance cheap train tickets but they have bus and budget air choices.And of course there are some good budget hotel chains as well, such as Travelodge and the Tune Hotel
Have fun and keep watching the site!