I was looking for the exact same thing and found a plethora of PHP frameworks. I'll tell you right now -- I did not find anything close to the goodness that is jQuery.
I ended up choosing Kohana, which I consider a mistake. CodeIgniter would have been a better decision, but I still feel that you're adding a lot of extra data.
For instance, jQuery adds on a lot of extra functionality to anything you're doing right now, it doesn't demand you fit certain specifications. It gives you extra functionality, not restricting anything. PHP Frameworks, from my experience, make you fit certain things (such as MVC requirements), which can be both good and bad, but performance-wise you will have a decrease (I have not seen any PHP Framework that can come close to the performance I get out of my own programming).
I've heard as well that CodeIgniter has EXCELLENT documentation, where as Kohana, does not. I would go so far as saying it has TERRIBLE documentation. Kohana is made for PHP5 so I thought it would have more of a performance increase over CodeIgniter (which has functions made for both PHP4 AND PHP5) and saw many reviews of people who had moved from CodeIgniter to Kohana, and they said Kohana was better. I disagree, they also said to learn CodeIgniter first, then move (because of its good documentation, and it's very similar).
I ended up making my own framework which is much closer (in my mind, at least) to jQuery. I can send this to you if you like, just realize that it is in no-where near a "production" level. It is more based on the idea that it gives you tons random helpful functions and tools than on putting your coding in another structure. I combine it with a WordPress similar system, the routing and so on -- but that is separate. This is all in one folder and you include one file.
I look forward to seeing the other responses.