Sorry if I offend, but I hate, hate, hate stray and feral cats (and stray and feral dogs). They kill millions of wild birds a year and harass our chickens every day. I'm so sorry you had so many chickens killed by cats. Our shelters are full and they spay/neuter and release anyway, so there's not much that can be done about them where we live.Do not attract cats to your run. Your cat may be fine but most aren't. A few years back I had 3 get into my coop and they killed 41 out of 42of my layers for sport, one bite was taken out of all the kills.
Seconding the coop security suggestions. Our coop and run are totally covered by a shingle roof, corrugated steel roof, and hardware cloth in different areas and covered completely on all sides by chain link covered in hardware cloth. The hardware cloth is held together by wire and plastic zip ties and I check for any gaps almost every day.
Our predators are raccoons, opossums, hawks, dogs and cats. There are coyotes in the area, but they seem satisfied so far with rabbits and stray/backyard dogs and cats. We're looking into electric fences just to be on the safe side and lock up our flock every night after they free range during the day.