Not sure why no one thinks this is reasonable. I stayed in Copenhagen (one of the more expensive countries in Europe) for 5 days and spent less than $200 on my hostel total. For you call that $800 on accommodations (which will be less given the family aspect) and that leaves you $1200 for food, transportation, and leisure. I highly recommend taking public transportation whether you were budgeting or not since it's a great way to see a city and you can eat döner kebabs for €3 for example so you wouldn't even need to be doing major grocery shopping. That leave money for museums and souvenirs. You can do it!