Well, to start with, I feel your pain. I'm the introverted, easily-exhausted-by-social-interaction mom of a chatty, highly emotional extrovert. It's so tiring! He wears me out!
Still, in my experience, I'd say that part of the problem here is your approach. I don't think it's your responsibility to make your daughter happy. It's your job to keep her healthy, fed, safe and educated. Happiness is her problem, and she's old enough to start finding ways to keep herself busy without you, assuming there are no extenuating circumstances that make this really impossible for her.
I'd be requiring alone time if you're not already. Start with shorter periods of time and gradually extend them, and do it every day. This is good for her to learn and will help you to be more patient with her the rest of the day. Do make sure that she gets plenty of your full attention before you start the alone time so that she has her batteries recharged before she has to be alone. I'd even explain this to her in this way: that you get worn out by being with people and she seems to get worn out by being alone. So you'll have to find a middle ground that works for both of you. It's not personal. You're just different, and it's important that you both get your needs met. My 5yo extrovert understands that and is MUCH easier to deal with since I started explaining it to him that way.
I think you also might need to find some help. My life improved greatly when I found a few hours each week without my kids. Is there a co-op or homeschool group available that would allow you to drop your kids off there? Someone you can leave the kids with for a few hours once a week or so? If not a spouse, then a sitter? A friend you can trade childcare with? That might be a good solution since it would mean regular playdates for your daughter and time alone for you sometimes.
As for friends, we've had good luck with kids who are a year or two older. There's often a better connection for our DSs (both the introvert and the extrovert) than with kids their own age. If there's a gifted support group in your area, that might be a good place to look for same-age peers.
Shared interests really help. Age matters a lot less if the kids are talking about the same thing. Helping kids get involved in some pop culture stuff that lines up with their interests can help make it easier for them to find things to bond over with other kids. Look for these things. They're different for boys than for girls a lot of the time, so I can't be much help here with specifics, but Pokemon, Legos and Wii have worked well for my boys.
Also consider finding boys for friends. A lot of times gifted girls hit it off better with boys for whatever reason. My best friend growing up was a guy, and decades later, we're still close. My boys have had some good friendships with smart girls.
HTH! Talk more if it will help!
