Wow, I read the whole thread. That sounds like a terrible place to work. I'd honestly call it quits.
If that is not an option, I would try to maybe break the children into two groups. Your co-teacher can bring one group of 4 to one center and you can bring the others to another center. If you absolutely have to have the whole group in one center for 15 minutes (I wouldn't even think about 30...Pre-k kids can hardly do that) I would try to do a short 5 minute introduction song or Circle Time to the activities in the center. Do lots of movement activities involving the items in the center for maybe another five minutes. For the last five minutes, perhaps introduce a fun, new item that you will add to the center.
The way you are describing your director's wants seems totally unreasonable, especially for one year old's. That is not DA...what is the admin's logic? Would she be willing to come in and demonstrate what she means?