We always had cats and dogs when we were growing up and basically they all got on together, sleeping together etc. We were just really firm (loud yells) if the dogs showed any signs of possibly harming the cats. Some of the cats used to tease the dogs (boxers) by ambushing them etc, but it was all just having fun. I remember a visitor being appalled at seeing our little burmese all slobbery from the boxers, but what he didn't see was that the naughty cat was always the one instigating the trouble.
It would have been better I think if you could have started the cats and dogs being together from when the dogs were little, but in your shoes now, I think I would be trying to get it into the dogs and cats heads that you are all family. At the moment the cats and dogs do not interact and the running through the dogs rooms is more like a prey action than anything). For instance I might sit on a beanbag with the old cat say reading a book and stroke the cat, and also stroke the dog. It might be best to do one dog at a time so that they don't egg each other on. If the dog wants to sniff the cat I would certainly let him - and show the cat that she is quite safe (and not let the dog annoy her). In the unlikely event that the dog wants to do anything to the cat while with you, I would be really firm with the dog (loud yell no).
You need to take your time with the introductions etc, in part because dogs and cats have a different language (tail wagging cat is unhappy, tail wagging dog usually happy, tail up cat might be a warning, tail up dog just expresses confidence etc).
You need in my opinion to spend the time to change the dog's mindset, which i think is quite doable if you try.
Best of luck!