Everyone is different. For me, the London Eye was not only not a "must," it was a "must avoid!" So, if you and your kids want to do it, do it; if you skip it, there are certainly plenty of other ways to spend the time and money.
I did enjoy the view from the top of the Tate Modern on Saturday night (free). The art in the Tate Modern itself was much more of a mixed bag for me.
I enjoyed the Brick Lane Market on Sunday, and I think you and the kids would too. Nearby is the Old Spitalfields Market, now with lots of trendy eateries and shopping. Both were hopping and lots of fun; if you or the kids want a break from historical stuff, this is a great glimpse into modern London.
In addition to the Tate Modern and the Sky Garden (which I didn't do), there are free views from the top of the One New Change mall. But in general, London's skyline isn't as impressive as you'd think a world class city's would be; that's why I wouldn't focus on seeing it, but rather the spectacular specific attractions themselves.
The National Gallery, for me, is fabulous; I go every time I'm in London. Room after room of masterpieces; it's not one of those museums with only a few good pieces and the rest second-rate stuff. It's the only place where a Vermeer could get ignored (when I went last time, I was the only one in front of them!) So it is important to be choosy, and focus on which periods, artists, and/or pieces you want to see, to avoid overload.