There are several related plug-ins for all the major CMSes. WPtouch is popular and has a proprietary version that costs $39. Joomla users can turn to TapTheme for similar skins that work with iPads as well.
All these approaches offer more control than any mobile repackaging site. The code is generally open source, and modifications are both often possible and usually feasible. When the modules are well-maintained, this approach can offer most of the advantages of a hosted mobile repackaging site with the control of owning the server. But if the module developers get called onto other projects, the work falls onto your shoulders.
Sencha's professional touch is apparent. There are more features, more panels, more widgets, and more design options than in open source options. The documentation is also much better than many of the open source projects, but it is still not as deep as I would like; on several occasions, I found it was missing or just based on an earlier version of the code.
The simplest approach is to place the package on a Web server and let the Internet do what it does best. If the HTML and CSS code is well-designed, the result will be almost indistinguishable from an installed app.