Congratulations on your new baby! I can relate to the challenge of nursing a newborn and trying to play with and take care of other children at the same time. I am still nursing my 9 month old son, and my daughters are 2 and 4. I can tell you that the first few weeks of nursing are the most intense. Your newborn will settle into a more predictable routine soon.
I admire your commitment to play with your kids, particularly only 1 week post-partum!!!!! Nursing isn't for everyone, but I have found it to be extremely rewarding. Even though you are an experienced M., it sounds like you are relatively new to nursing, so pumping and nursing sessions are still pretty new. Give it a little time to see how things settle out. Your newborn is nursing so much because she needs time to bond with you and that is her way to ask for your nurturing. I know her demands may feel minor compared to the more dramatic demands of your older children, but they are just as important. Even if you were formula feeding, it is likely that she would want this nurturing from you.
Anyway, when my son was really little and could easily nurse without being distracted by his sisters, I found I could nurse using a boppy pillow and play card games, do puzzles, or build block towers with my older kids. This takes some practice, but it can totally be done. I also admit that I relied on TV and movies, which we would watch together (and still do).
This is just the beginning of a very long relationship between your children, so be careful not to let your kids start to feel resentful of the baby for taking you away from them. It sounds like you are doing an amazing job and that you are a very conscientious M.. Give yourself some time to bond and nurture your baby, and after you settle into nursing, if you decide that it isn't for you, then you can move on knowing you gave it a good effort. By the way, I wasn't able to produce much milk through a breast pump until my babies had started sleeping for a couple hours for naps. Until that time, I didn't have enough time to build up a supply.
Good luck to you!!!