Might I suggest London for your anniversary trip? I think you can definetly
fit that into your budget, see tons of spectacular things, walk to your heart's content, eat well, be romantic, etc.... London is my favorite city: so full of history, art, and life... so many things to see and do... and easy to get around on a budget.
There should be great flight prices available to you, and once you are there you can buy a Tube pass to get around economically and conveniently. I don't think you should even think that you are doing it on the cheap. We travel quite a lot on a budget of 3000 USD per trips of seven to ten days, and I like going to Europe best.
We are going on a seven day trip to Paris (six nights) and flight and room will run about 2000 USD. The rest is for eating and sightseeing, and I think it will
be plenty. I vote for any city in Europe...you can do it within your budget.
I teach an adult education class called Europe Cheap With Style and Comfort.
I know you can have a great trip on your budget, and still be comfortable.