I have very little experience with CakePHP so my answer is going to be about CodeIgniter. I played with CakePHP for about a day and that was almost two years ago. In my opinion it will probably be easier to integrate with CodeIgniter although someone more experienced with CakePHP might prove me wrong.
Here is the approach I would take. I have never done this, but it seems like a logical way to approach the problem. I suppose this approach would also work with CakePHP.
First, start with a fresh CodeIgniter install using the latest version.
Next, create controllers and actions (controller methods) that mirror the current structure of the application. For example, if you had a page with the URL http://example.com/users/view you would create a Users controller with a view() method.
Next, create view files for each of the current files of the application and load them via the appropriate controller methods. The goal here is to get the application working using CodeIgniter's routing system although at this point you won't be utilizing any models, libraries, or helpers.
Once you have the application sitting on top of CodeIgniter, start refactoring it to fit into the MVC pattern. Pull out application logic (queries, form handling, etc...) from the view files and place them into the controllers. Keep all presentation logic and HTML in the views.
Next, refactor the controllers. This is where it gets tricky because controller code can be placed into models, libraries, or other controller methods. A good starting point would be to take all of the queries and put them into appropriate models. Compare your controllers and see if there is any code duplication. That is a good sign that you should remove it from the controller and place it elsewhere. Unfortunately I can't really tell you where because it differs in each situation.
Continue refactoring your application until you have it in a workable state that you are pleased with...
Hopefully this helps. I certainly missed some critical steps such as setting up and configuring CodeIgniter but if you're serious about doing this I would highly recommend reading through the CodeIgniter User Guide to get a good idea about how it works. You should also get familiar with MVC (model-view-controller) if you aren't already.
There's not really a one size fits all solution here but hopefully I've given you some ideas or at least a starting point to jump off of. If you have any questions or are a little confused drop a comment below and I'll get back to you.