It can be difficult getting to know Danes as they tend to be quite family oriented, rather than 'friend' oriented if you see what I mean.
Do you speak Danish at all? That is a big help as in spite of the generally good english skills here, there is definitely still a bit of a language barrier, and the english classes the kommune offers are a good way to meet other foreigners in a similar situation.
Have you tried contacting the kommune regarding children's activities? Here there are loads of things on offer, but you have to go out and actively ask about them, they're not generally well advertised (and not usually in English either!).
It may also not be that useful to you but have you tried looking at Mødrehjælpen?
www.moedrehjaelpen.dk/default.aspx
They are sort of an equivalent to the NCT, though not nearly as big and with an emphasis on families in financial difficulties but they have local branches who organise local events for all families. Several of the local branches run cafe butikken where you can meet up for a coffee, buy second hand stuff etc etc. There's usually someone around who doesn't mind chatting in English if you're not up for it in Danish. So, it might be worth looking into.
I can't imagine living in the back of beyond of rural Denmark, so I can quite see why you may be finding it tough to adapt and integrate, how long are you planning on staying?