We definitely have a lot in common single dad. And I congratulate you on keeping half custody, half placement. God knows that is enough of a triumph right there.
Anyways, I was/am in the same situation, though my daughter is 6 now, when her mother and I separated, she was 2. And then followed a year and a half of terrible, custody-court warfare. Luckily thats all over now and I also have pretty much half custody and half placement.
Its true that the age your daughter is, is probably the hardest to find things to do, and its taken me a little time in trying to remember what kind of stuff my daughter and I used to do back then.
A great idea is going to the park. I know that doesn't seem like much but its a great way to really soak up the nice weather, which is a joy in itself, while at the same time letting your daughter run wild and expend all her energy so that when bedtime comes, she'll be out in minutes. Also, its a good way to meet other parents and sort of network, since other parents are always sitting around, watching their kids too. I met a mother at the park we always went to, who had two daughters around my daughters age and they ended up being great friends and having playdates with each other, and they are still great friends to this day.
I don't know if you have many yoga studios around where you live, but the ones here usually have a class for parents and children from like, 2-4 years in age, where you can do yoga together. that's a great way to bond, and she'll learn bodily discipline as well, and how to concentrate.
Dance studios also usually offer a class for girls her age. You probably won't get to participate in that, but you'll get to watch from the side and beam with pride!

Buy puzzles. And sit down with her and help her do them. Considering her age, I would try puzzles with 30-50 pieces. I know that seems like alot but trust me, if you help her, she'll be able to do them, and it will occupy her for an hour or so, which will also be great bonding time.
Okay, I could go on forever, so I'll make the rest of my suggestions short. Play-doh or modeling clay. Make animals together, or like my daughter and I used to do, create all new kinds of monsters, by combining parts of animals that already exist. Or just buy a coloring book and color together. Put some music on and as you color, engage her in conversation. Some of the best times I've had with my daughter were just quiet nights at home doing things like that. Once the conversation starts rolling, she'll surprise you every time with some small flash of intelligence or cleverness you probably had no idea she had yet. if i can think of more I'll post again. Hope that helped.