Yes, I have done some research! I have an indoor pet duck so she gets a lot of attention and pets. The concern is that she heads into mating position when her back is being pet because she's aroused. This is what leads up to going egg bound. If her reproductive needs aren't satisfied, she will go egg bound.
I have learned a few good things to do. The very first and most important is prevention. This includes, take it so easy on the back pets and if she's in mating position then stand her up or walk away. The next good thing to have on hand is a great link from Metzer. I keep this link bookmarked on my web browser.
https://www.metzerfarms.com/EggBound.cfm?affiliate=undef&CustID=23327665
Like in your case, the duck just ended up egg bound naturally and it is too late for prevention. This is when I would go through many steps to try to get the egg out. There's many different methods and try them all because you never know what will work for your duck. Here's some idea's and info but always research to see what has worked for different people.
https://www.shagbarkbantams.com/egg-binding/
https://www.backyardchickens.com/articles/egg-binding-symptoms-treatment-and-prevention.66978/
http://www.duckdvm.com/condition/egg-binding
If egg is passed, I would consider lupron shots. If egg won't pass, I would consider hysterectomy.
Of everything I learned, there is no easy option and the best possible option should be discussed with the vet. It really depends on breed, age and vitals. The lupron is expensive long term and the hysterectomy is expensive short term.
I wish you very good luck, please do keep us updated. This is a subject I would love to learn more on!