I made a quick check at DSB udland and there I can see 3 connections but only 1 where you reach Lille the same day. I hope I understood correctly that you need a normal ticket. The price is for one adult older than 26.

https://travel.b-europe.com/dsb-rail/en/booking#Outbound

Cheapest is what takes an awful more long time-as DE=Germany offers the 9€ monthly ticket for local trains only=can use from DK till FR border.

You should have bought a global pass, as this would be valid all over EUR, incl 1 trip OUT and at the and back IN own country.

Also the route via BRU is more expensive-you can also reach FR direct from DE-in that case check what DB offers as SPARpreis (fixed advance) from Khn till f.e. Saarbrucken-or Offenburg, then buy cheap local single till first stop in FR and use pass from there.

Yes one person normal ticket from Bruxelles to Lille. From there the French Pass would be valid.

I saw the DB 9 Euro Ticket includes only Regional trains so it would take a long time to arrive to France. Say I go to Hamburg from Copenhagen and from there continue the Journey. True, a Global Pass would have been the wiser choice, Idk why I had them remembered much more expensive than they actually are. 

What's the best connection with a Global Pass? Maybe I buy it for this Journey now since the France one is not activated yet and I could use it another time. 

Depends what you'd call "best”.

The timetable AnnaB posted is OK, just that you'd need a 20€ reservation for Brussels - Lille (which is around the same price as a normal ticket...)

Instead, you could also do this:

You only need a reservation between Kopenhagen and Hamburg, which you can buy via DSB. When arriving in Cologne, do check if the ICE will run. If not, continue to Aachen ASAP and look into options via Welkenraedt. Also, install the DB Navigator app to be able to see real time information.

Thanks for your help. Much appreciated!

What's the best connection with a Global Pass? Maybe I buy it for this Journey now since the France one is not activated yet and I could use it another time. 

Depends what you'd call "best”.

The timetable AnnaB posted is OK, just that you'd need a 20€ reservation for Brussels - Lille (which is around the same price as a normal ticket...)

Instead, you could also do this:

You only need a reservation between Kopenhagen and Hamburg, which you can buy via DSB. When arriving in Cologne, do check if the ICE will run. If not, continue to Aachen ASAP and look into options via Welkenraedt. Also, install the DB Navigator app to be able to see real time information.

If I have a Global Pass but start my journey in another country (Copenhagen/Denmark) than my home country (Austria), do I still get to benefit from the option to have a trip back to my country or is it not included anymore in that case?

Thank you very much for your help!

Is it better to make a reservation for the other trains as well or why would the ICE from Cologne not run? 

With a Global Pass you always have the possibility to use 2 travel days in your home country. It doesn't matter when you use these days. They can be in the beginning and the end of your trip, in the middle or you can use both at the end of your trip. 

Deutsche Bahn has problems with their ICE trains that can run to Belgium and the Netherlands, so some time the train will not run or run only on a part of the route. On this route there are also Thalys trains, but they need a reservation (it costs about 20-30€) or via Aachen you travel with local or IC trains. 

For the reservations for the other trains, visit for Germany bahn.com you can see there how full will be the train will be.