Welcome new user!
- It is utterly, totally, completely inconceivable you can fly to the US with an expired passport.
Nothing more need be said on that.
Unfortunately, if you look up "non-starter" in a dictionary you will see a photo of someone with an expired passport, trying to fly to the US.
Phrases like "beyond the bounds of reality", "beyond conception", "staggeringly ridiculous", "this is funny, are you joking" were invented for this, sadly.
It's weird you were not able to get an ESTA with your ordinary Danish passport but, so be it. Normally you just don't mention any other passports. But what's done is done.
I'm not Danish, and nor do I hold nor would I ever hold a US passport, but as a general rule it is extremely hard to urgently renew a passport when you are not in the country in question. It's very, very, easy to renew a US passport, within hours, if you are in the USA. (Newsflash: governments love you giving them money. :/ ) I fear it will be difficult/impossible to renew your US passport urgently when you are not in the USA.
In true, honest-to-God emergencies (involving nuclear plants, politicians hiding money, etc) there may be a route, but I fear there's no chance this would apply here. Sorry for the downer news.