Take the lily out of the water as soon as possible, and change the beta's water as they do not do well in water with rotting vegetation. Trim the rotten roots of the lily and replant it in soil that is moist, but not soppy.
Often times the peace lily is getting too much water and therefore rots in these vases. My suggestion would be to throw away the peace lily, clean out the vase and then place the beta fish back in.
It sounds like the plant is getting too much water. I would suggest repotting it in a well draining soil and letting the soil dry out between waterings. You shouldn't be watering more than once a week. I would keep the Beta in a seperate vase.
You should remove the peace lily from the vase and plant it in a pot with soil. I've seen these vases at a lot of different places and the plant does not seem to survive long term. The fish will be fine without the plant.
