Yeah!!!! They're no longer monolinguals!
Remember that speaking is the difficult skill. Understanding a language is not very useful. Now they have proven that they can do the more difficult part; speaking, grab it and run with it.
You may not consider answering in Danish a "problem". It's not really because they only need one language to get by in life. But if you want them to speak English, you need to guide them. Think of it like table manners or sharing. They may not do it automatically. It's easier to do things in the better language for anyone!
Just be really positive if they use English with you. Be poky and slow with any commands made in Danish. Read to them in English and test them on vocabulary. Bilingual children need lots of vocabulary input to keep speaking the weaker languages. Without the words they need, they will resort to Danish so make sure they can say what they need to.
Play with it. Make it fun. I ask numbers, months of the year, days of the week, etc. in the car in the three languages my children speak. Name things around the house and outside the house. Be sure to use English when out in public so that they can see it's not something to hide or be embarrassed about.
Be zen with the fact that their Danish will be better than their English. This doesn't mean that they can't keep the English up and be able to communicate with their American relatives easily. Don't give up on this project because of that fact. Now that they're a little older, they might be happy to put their skills to use so find some opportunities to use it.
I got out a map one time and showed them all the countries that spoke English. I then compared it to where French is spoken. Wow, their eyes went all big! Now it's this big, fun, novel thing to them.
Speaking another language is such an advantage in life. Now that you're over the big hurdle, it would be a shame if they slip back into their all-Danish world. My children actually asked once why I always spoke English to them. I explained that it was like an animal who has to be fed and watered or it will die. They kind of thought it was something they could just do and it would stick.
You have two big advantages over me. I live in a country where few speak English and my dh can't speak it either. In Denmark, you're assured that they will learn it at school and probably your partner speaks conversational English.
My kids have made huge leaps forward on vacations. English all of sudden becomes something very real to them.
Just keep it up... Don't force or even insist but with a positive attitude, you can get them actually speaking English and being bilingual.