Since you asked the simple question: How?
Obviously, it can be done. The question is ... is it worth the trouble?
Let's cover the basics.
Once you have the landing page built, you will need to copy the source code. To do this go into the file manager or FTP and edit the file, copy the HTML to note pad and save the file as - landingpage.html
Move the images, flash files, and .html files to the new website. Make sure that any links to the menu are edited.
*** VERY CRITICAL *** Test all links when you move the page. Bad links can kill the performance of your website.
If you are using flash - you will need to make sure that you also copy the related flash file, usually .swf or .flv.
Look for the line in the code that references the flash file and change the path to the web root for the site you are moving it to.
It is unlikely that there is a database file involved. If there is, it would be easier to build the entire site in the site builder tool that you are using and then export all files in the web root and look for the connection string (usually config.php, connect.php, includes.php) - This is where you need to decide if it is worth the effort. You need to create the database, create the user for the database, add the user to the database, import the data and finally set up the connection to the database ... that is a lot of effort for a freebie ... and twenty years of experience has taught me that the people who are hardest to satisfy are the people who pay a premium for your work and those who are given a huge discount. So take my advise and if a database is involved, recommend that they let you build it using the web builder on their site. They can change passwords when you are done.