Okay well, at least the good news is that your shifter ain't broke literally. Now it's just a case of getting it assigned in game..
Open your LFS\Data\Script directory, and find a file called "sequential.lfs" in there. Open it using Notepad.
In there, replace the content with the following:-
// Set sequential buttons
/button 12 shift_up
/button 13 shift_down
In there, you can see "button 12" and "button 13". These may not be correct for your DFP. They're correct for mine, but it can vary depending on which USB port you connect to, at least it can for me!
To find out which are the correct buttons for you, open LFS, go to Options and then click on Controller. While you pull your gearstick towards you, a number will appear on the screen. This is the number for shift_up (12 in my config). Push the stick forwards and the number that appears is the one for shift_down.
Correct the numbers in the sequential.lfs file, save the file and your gearstick should then be configured for cars with sequential gearboxes.