It depends on what she is after, if it is a material thing, that I would advise that this is an almost impossible challenge, especially if she is rather fussy.
Option 1 - A day just for the 2 of you
What I would want (and I say this as a fussy female) is for my partner to take me out somewhere for the day that incorporates some form of shopping, this way you can suprise here with a location or event and a great day, and she can also get just what she wanted in terms of an actual gift.
Take notes of what she buys for next year, and also look what she buys as gifts for other women. When I am shopping for gifts, I tend to pick things that I like, as there is a good chance the recipient will like them too.
It is important to note however that with this option, taking children along isnt ideal. Birthdays are meant to be all about her, and that just isnt possible with childeren around as they will always be the priority for both of you, as they can also be unpredicatble, the day can easily be spoilt with one spilled drink.
Option 2 - A family day
Having said this, from what you mentioned with the baking, while that sounds like a very sweet thing to do, there were 2 issues, firstly it sounds like she would have rather been out of the house and secondly she would have liked to have been more involed in enjoying the activity. One idea might be to go somewhere with a large indoor swimming pool and a relative that can watch the children while they play.
The reason I say this is because she will get to enjoy being close to her childeren in an environment where there isnt much they can do to misbehave (i dont have kids so ignore me if thats a pack of lies). She can also enjoy some quality time with you away from them if there is a hut tub area, and you can both act like big kids on the slides. It will tire them out ready for an early night ready for you to bring out all that romantic good stuff.
Option 3 - A party
What better way to make her feel special than with a room full of family & friends! one of them is bound to bring her ea gift she likes, if not, ask guests for donations toward a larger present that you have in mind, for example a holiday or romatic get away.
Option 4 - A day away from all of you
Maybe she doesnt get out much and would like a little break with some of her friends? you can look after the kids all day & get the house spotless for her return. Get the kids to make her something!
Option 5 - Do what she does for you
Think about the effort she puts into your birthday? maybe tell her nothing and see how she reponds to the situation. Do for her what ever she does for you, as again with the gift buying, it could be that what she does for you is something she would also like herself.
Option 6 - Be sneaky
Show her this post, or post something similar anonymously, get her to tell you what she thing the man in this situation should do, or ask her friends & family!
Option 7 - Nothing
It has been said in other answers, but ultimatly she should appreciate the thought and effort that you have put into planning the day. Doing nothing is a risky move, but if she is as ungrateful as you are making her out to be, then maybe some tough love with a dose of reality is what she needs to be reminded how lucky she is, that you care enough to post a thread online about it.
One final thought, make sure she feels appreciated all year round :) & if there are any underlying issues in your relationship, then nothing you can do will make her happy for just on day. But fixing a relationship sounds like a whole new thread to me!
Good Luck & Godspeed!