Hi NathanVaughan.
There is really one great place which needs mentioning in Copenhagen.
And that is Casino Copenhagen. Largest Casino in Northern Europe and with best facilities.
The hotel is not on the cheap side. So you might wanna find one nearby instead.
Or else can recommend you this place.
Radisson Blu Scandinavia Hotel, Copenhagen (Casino Copenhagen)
Enjoy your time there. And just remember it is quite expensive there
http://www.radissonblu.com/scandinaviahotel-copenhagen/casino