Hello,

bake them in an oven as long you can and pray they will set

~~~Everytime I hear sound of demoulding part I have eargasm~~~

If it will set or not really depends what the "problem" is. 

Did you use the correct ratio of resin to hardener?
Properly Mixed?
Temperature?
Humidity?
Contaminants?

They are the obvious things that can cause issues.  If the resin and harderner ratio is massively out then you will struggle, as will you if the mixing was poor or there were inhibiting contaminants (rare with epoxies).

However, temperature and humidity can be solved in most cases.  Heat often solves both.  Uncured epoxy is fairly hydoscopic so absorbs moisture so often lower temperatures cause 2 problems, by delaying the cure due to temperature, thus giving the opportunity for moisture to be absorbed  - especially in sheds/garages/outbuildings. If the resin hasn't cured too much you can sometimes with heat drive the moisture out and get away fairly lightly with just needing a bit of sanding and polishing to restore the surface finish. 

Warren Penalver
Easy Composites / Carbon Mods - Technical Support Assistant

I haven't measured the room temperature unfortunately but it was definitely above 10 degrees C 

Would a heat gun work? I tried it today for a couple of minutes on one of the parts and it seemed to turn the tacky parts more liquid-ish. Should I continue to apply the heat gun?

Can you get hold of an electric blanket,

wrap it up and cover with a duvet, should sort it.

Did you mix the resin well? Because this could be a problem.. If it is too cold the resin will dry slower.. You can buy a infrared lamp (used for example for little animals or plants).. It works well for me and the parts are ready in 2-3 hours.. Maybe you can try but if you mixed bad the resin is a problem for the structure in my opinion.. 

I am confident the ratio was correct however I may have not mixed the epoxy and hardener very well despite leaving them to set and degas for about 10 minutes. The only heating I tried was heating with a heat gun for about 5 minutes and stopped. The carbon layers were getting very hot and the epoxy was turning from tacky to runny, I stopped as I wasnt confident if I am doing the right thing. Would you recommend I heat it for longer periods?

Heat guns are extremely hot, up to 250 degrees depending on how powerful they are and what setting you've got it on. Way too hot regardless, you want about 50 degrees - 60 degrees tops, even 40 would do. And for a period of a few hours, ideally 5 or 6.