I'm rather impressed with Dell for taking responsibility for this situation and making it right so quickly! I also like that they send you the replacement system FIRST, then you send the defective back to them. Granted, the broken one isn't of any real use any more. But it will be reassuring to have a brand new, operational unit in your hands before you send yours back to Dell! It's sort of a good faith effort on their part.
As for the data/programs on the hard drive:
IF the replacement system has the same size hard drive as your current one (and the same version of Windows), the easiest solution is to simply swap out the drives between the two laptops. Don't panic if you've never gone near a laptop with a screwdriver before! It's as simple as removing two screws to release each drive and sliding it out of the slot, then doing the reverse on the other laptop. Dell provides great instructions on how to do it as follows-
That link is actually for the entire Service Manual for the 1750 (Inspiron 17), just click on 'Hard Drive' to see the specific directions. Basically, you just flip the laptop over (remove the battery and unplug it before you do anything else) and locate the two screws holding the hard drive in place. On your system, the hard drive is located at the front of the the left-hand side (in front of the USB/network/VGA ports). It has a plastic bezel on the front edge that you would normally remove, but you can just swap it between systems since they're physically identical. It's so simple.
You were very wise to create a backup on an external hard drive, btw! If you don't swap the drives out, you'll need to connect the Seagate external drive to your new laptop via USB and manually copy any files onto the new system.
An added benefit of physically swapping the drives is that you'll be sending back a hard drive with NONE of your personal information on it! Your identity and data will be much safe that way! =)