Cycling is growing bacteria in the tank which remove certain types of fish waste from the water. What you can see is poop, but fish also have their version of urine which you can't see, ammonia. Without these bacteria, ammonia builds up in the water and harms the fish. In a tiny jar like this, ammonia will build up quickly. Some bacteria will grow on the walls of the jar but the washing up liquid and the chlorine in the tap water will kill them off at every water change. So the poor betta is basically living in toxic water. Cycling can be done in a few ways, but if the owner goes back to washing the tank and using untreated tap water, it will quickly be uncycled again.
The best you can do for now is to do the water chnages, and when she gets home, try to convince her that she needs a proper tank for him, one with a heater to keep him warm and a filter as home for the bacteria that need to grow. That she needs to do water chnages using warmed tap water that's had water conditioner added. And that she must stop using washing up liquid to clean the tank.