I found the perfect framework. It's fast, easy-to-use, and extremely flexible.
The framework I decided on is Kivy, a relatively-new framework programmed in Python.
Before settling on Kivy, I took a close look at Corona SDK, Gideros, and Titanium Appcelerator. I also looked briefly at Phonegap and Sencha Touch before discounting them for the purposes of my app. Corona and Gideros are both appealing because they are relatively compact, perform. well, and programmed using Lua -- a language I've come to appreciate over the past few months. However, both of these frameworks are primarily directed at game developers, particularly Gideros. Corona SDK has started marketing their take on business apps to enterprise clients, but this option is quite expensive. Most importantly for me, Corona SDK only runs on Windows and OSX, and I'm developing on Linux. Gideros appears to be well-designed, but is even more oriented toward game developers than Corona. So both the Lua-based frameworks were out.
