You say you've thought about playgroups but have a hard time in groups. Is the playgroup for you, or for your daughter? It might be more your issue than your daughter's. Believe me, I'm not all that jazzed about a lot of the other moms out there, and I'm sure the feeling is mutual. Sometimes, though, I just have to bite the bullet if it's for my kiddo's sake.
One thing that helps me when inviting moms/kids is to pick a specific time and date. Whenever I send an email to another mom and I say, "Oh, we should get our kids together at my house sometime for blahblahblah", I never hear back from the moms and, truth be told, vice versa. When I say, "Pump It Up is having their winter break discount on Thursday. I'm taking E. over at 12:30 if you want to meet us there," I get an answer. Even if it's "no", it opens the door for a concrete date later on.
Oh, and about those firm dates and times? Check out Park District events to attend. They're usually free or very affordable. Good luck!