We have ants in our house, and nothing I've tried so far has been sufficiently successful. Now I'm considering using ant traps, but only under the kitchen cabinets behind the plinth, a place where neither my cats nor my kids (a baby and a toddler) would ever be able to reach them (no chance of them getting in there). Would that be safe? Are there other aspects we'd need to consider?
When you need to leave food out for longer periods of time, you can put the food bowl in a pan of water so that the ants can't reach it. Of course the pan should be small enough that the cats can eat without getting their paws wet.
The easiest way to get rid of the ants is to remove their food source, which you have done. In addition, you could inject some sort of ant killer behind the baseboards and in other crevices. Borax is probably a good choice; it is not safe for cats in quantity (e.g. to use in a litter box), but minute amounts should not cause problems. Some people report success using soapy water, vinegar, or even catnip!
There are also products that contain ant bait/poison inside a plastic case, with openings big enough for an ant to get through but not a cat's paw. As I recall, these look rather like those round stick-on air freshener thingies. Assuming your cats are not in the habit of eating ants, these should work well and be safe.