A $4,000 budget will buy very nice hotel accommodation in many wonderful places in Eastern Europe. In some cases the best hotel in town, but definitely good accommodation virtually anywhere. Budapest, Prague, Ljubljana, Gdansk, Krakow, Warsaw, etc... are all lovely authentic places that are absolutely worth visiting, even if one has an unlimited budget.

Paris and London in Particular are exorbitantly expensive for hotel accommodation. Unfortunately this has only exacerbated in recent decades.

However, as eleven nights is a relatively long stay in one place, I think you could definitely get a fantastic vacation rental apartment, or a lovely boutique hotel in certain places.

By the way, how many Marriott (formerly Starwood) points do you have? If you have sufficient points, you could get a few nights at an hotel such as Prince de Galles, St. Regis Rome, or Park Tower Knightsbridge, etc... if you did want to have a nice hotel stay as well.

There is no way in the world that you are going to get a top hotel in Rome, Paris, or London for anything close to $4k for eleven days. But an out of the box solution like I mention above should be totally possible.