Some thoughts, with the caveat that it's very not my field.
> cannot keep up
Different approaches can have very different demands. Some parents it seems, write out lesson plans -- lots of effort. Or instead, you can support, and mentor, and be an example, and otherwise "get out of the way" -- smaller and more flexible effort.
My fuzzy recollection is, one reason families without a higher-ed background, have kids that achieve less, is that the parents are more likely to believe, and teach their kids, that education is something that school is going to do for you. To you. Rather than something you have to create for yourself. With school sometimes helping, and often not. So one hears of community college students, protesting in frustration "I've paid you, so why haven't you successfully taught me yet?" Interestingly, a similar issue shows up with Ivy League students, who have had excellent teachers presenting knowledge well, and now lack the motivation and skills, to themselves grapple with a body of knowledge, and wrestle understanding from it. "They expect me to spoon feed them" an instructor said. So the reason for mentioning this... perhaps thinking of yourself in a "supportive" role, rather than a "school teacher"-ish role, might be useful?
Also... I knew a toddler, with their own toddler-high kitchen snack shelf. And they learned to pour cheerios, and water. More or less neatly. And two years later, that was all gone, and the parent was doing everything themselves. Almost passive "spoon feeding". So both the learning had stopped, and the parent was more busy. When young, so much can be a "learning opportunity", that "here's a way to save time", can at least sometimes be made into one. For a double win.
> by the time she starts school, she's going to be so far ahead that she won't be able to relate to the other kids
Some schools have multiple grades in the same classroom. Which can help.
> No idea what to do.
I've been told there are very helpful discussion mailinglists for parents of "gifted children". With supportive communities.
And I'd be unsurprised if some programs like http://cty.jhu.edu/ are used to getting "heellpp!!" emails, and sending out a resources packet. Or have it on the website? See comment below by dsjoerg.
> hang out with kids like herself and do what kids do
Just brainstorming... perhaps introduce the concept that "different people are different, and good for different things"? Grandma with walker can't play tag in the park, but can read stories. Some adults like talking with kids, and some don't. The librarian can talk about books, but can't make you sandwich, or show you how to do it. Some dogs you can pet, others only say hi to. Some kids do and like talking about activity X, and others don't care about X but like Y. Which is frustrating when you want to do something particular, and the person at hand doesn't. But looking for common ground is an important life skill. As illustrated by other comments on this page.
Good luck. Definitely connect up with communities of parents facing similar issues.