I'd definitely recommend what ResumeMan said. If your ambient is only about 70 degrees, there's no doubt that the beer itself will be fermenting well into the 70s, and given the English yeast already gives off some natural esters, you would probably have a lot of odd flavors going on if you went that high of a fermentation.
Putting the bucket in another bucket with water and ice is a good way to keep temps down if you can. And remember, this is really only necessary during the active fermentation stage, so typically after a week you wouldn't need to maintain the cold water bath. But if that simply isn't an option, then a cleaner yeast would certainly help a bit at the higher temps.