Copying just the recovery partition to your external hard drive will not work.
You need to create a hard disk image of the entire hard drive
(all partitions) using disk-imaging software, and save that disk image file to your external drive.
Then buy and install a new hard drive inside your Inspiron.
Boot your Inspiron from the disk-imaging software's boot CD.
Plug in the external drive and write the disk image to the new hard drive with the disk-imaging software.
A selection of free disk-imaging software is here:http://www.techsupportalert.com/best-free-drive-imaging-program.htm