In order to get the Golden Floof Ball, you need to bring the regular Floof Ball on a normal mode run and keep it perfect condition until the very end (essentially doing a complete run without getting hit at all).
I got quite lucky and had a strong build already midway through the run. At that point I had so much DPS, even Shira at the end couldn't use her signature attack before I defeated her.
Build details (items and skills) at the end of the video.