Do you actually need to do anything with their existing site? Other than the content it contains?
Personally I would take the much easier option of playing around with Joomla. Find a pre-made template for it that would suit their site with a bit of treaking to make it unique. It removes a huge amount of work and hands them a fully featured and mature CMS free of charge.
If you decide to move forward with the project again you have to make it clear to them that the design and functionality of the site has to be your main focus and the actual site content will come later once they are happy with the look and function of the site.
If they agree that don't build anything. start by working on mockups of site designs in Photoshop or a similar program. Get then to look at the designs and they will have the opportunity to comment and change things at this stage. Once they are happy with the design get them to agree to sign that off.
The next stage you can then concentrate on making the site work. At this point you should be able to be left alone to work on it, and arrange a date when you will have something working to show them. You won't need to have the whole site conpleted and ready to go by the next meeting. Just a functional site that looks like the signed off design and a couple of pages of content to demo the functionality. You can utilise that to iron out any navigation and other UI issues they might have. And also typographical layout.
Once you have that stage complete and signed off you can then finish working on making the finished site fully working. They can then be shown how to use the site to add and edit content.
Normally around this stage the client will also have drafted rough edits of the most important pages to be included on the site. You will need to discuss site sections/heading for the navigation etc, then you can build the menu structure. They should then start sending you word documents containing content for different pages which you can start dropping into the site and formatting.
And then it's job done.