My son is 14 and an only child. I had him very young so by the time I even thought about having another, he still would have essentially grown up as an only child anyway. I have thoughts of guilt for him when he gets older and doesn't have anyone except us but I'm not going to have another child just for him or so that there will be someone to take care of us.
When he was younger he would say he wanted a brother but by the time he was about 10, he realized that being an only child has benefits. Enough visits to friends houses were enough to prove that he was just fine being an only child.
My brother is less than 2 years older than me and although we did play together when we were very young, as soon as we started school, we didn't have anything to do with each other and are not close to this day. We are just sooo different. I grew up always wishing I was an only-child b/c my brother had special-needs and it got so much more attention, deservedly so but as a youngster, I didn't really care why he got more attention.
Anyway, play dates are fine but I don't think they're a necessity at 6 if she's involved in other activities and school. In fact I think there are times when I went overboard with activities for my son. We've cut back to one sport, Scouts & church. Even now we have conflicts with overscheduling due to school work. My son loves to skateboard, ride his bike and play drums- activities that he can do whether he's by himself of with a friend. Obviously your daughter needs more supervision with some activities but you can help her find things she can do by herself and encourage those when you don't have time to play with her (puzzles, reading, hoola-hooping, jump rope, etc). Also encourage her to do house work or cooking with you. I'm a firm believer in kids learning to help around the house but it seems like only-children tend to be waited on more, I guess b/c there is only one to do stuff for.
An added bonus for us is that we have really good friends in our neighborhood who have younger children and our son has pretty much grown up with them (they're now 6 & 8 and we've been friends since the youngest was a baby). They're his 'substitute' brothers as we call them. He goes over and plays with them and helps them with their homework and when he's annoyed or out of patience, he can come home and be in peace.
And one last thing, my son doesn't have one best friend either. He's friends with almost everyone and it's almost always a different kid at our house every weekend (now that he's older we allow sleep overs and it's almost every weekend that someone is here). Most of the repeat invitees are more mature for their age and are willing to compromise by not playing video games for hours on end.
All that, don't feel guilty for your daughter being an only child. There are advantages and disadvantages to both situations and I'm sure you guys will make the most of it.