ES:
My son is five, also an only child, and still has days where he seems to 'need' me *all day*.
You are right, 45 minutes nonstop is a lot. And it is still seemingly not "enough" for them. Let's just acknowledge this right now: you are trying to fill up bottomless pit in some ways.
So time to back up and see what you are needing.
Know that toddlerhood is tough. Know that doing what you need to do is okay. At this age, I gave my son a small amount of playdough, and here's the 'bad mom' part.... When I gave him playdough, I gave him no more than I wanted him to eat. Because he often did. But I really needed him to be busy for that five minutes. So, I'd give him playdough and craft sticks to poke into it, and then I would hope for the best and try to get some dishes done. Luckily, I usually caught him before he got a couple of bites of the stuff in his mouth, but I realized a long time ago that if I wasn't willing to let a little folly happen, I'd burn myself out real quick.
All that to say, do what you need to do. Some kids at this age love a dishpan of water on the floor... if you need to keep your back to her, give her a soapy sponge and a cookie sheet and let her play with that on a towel on the floor.
I don't know if I shared this with you before, but I also had one bluegrass dvd that helped me get dinner made for a few months. My kid watched far more television as a toddler (not a horrid amount) than he has ever watched since. I needed it to keep me sane.
So, do what you must, use the tricks that work, and know that this will pass. I promise. It really does!