Daycare doesn't have to be all day every day. Our sitter had DS 2-3 days a week for maybe 3 or 4 hours at a stretch. It was enough time for me to get some of my part-time work in or to run an errand. He napped at home, and he got some of that social time and some new toys to explore at her house. Frankly, it sounds very much like what you're describing!
I'm not saying daycare is what you should/must do. Not at all. But I like to be sure people have all the facts before they rule something out, especially when it has worked for some of us in the past. If you find a good, attentive in-home provider, it might be just what you're looking for.
Is there a stay-at-home moms group around? They are all hooked in on classes and such. Go to the park and if you see a group of moms obviously there together, ask them. Or just check the yellow pages. If you know the names of the programs ("Kindermusik," "Baby Lingua," etc.) then you can even check the white pages. Local papers often have ads for them, too, especially the cheapy/free papers aimed at parents. Check kid-friendly restaurants, clothing stores, baby boutiques, etc. for these papers.
Even if you got her in, she's probably not going to get much in the way of academics in pre-K, frankly. Unless the pre-K is highly academic, and then there's a good chance the fit won't be good for your child, and wrong academics is worse than no academics at your DD's age. (Thanks, LMom!) If academics is what you're looking for, you're probably better off giving it to her yourself.
Hope that helps!
