external hard drive failure

How do I transfer the images on my backup external hard drive into Lightroom so that the program will recognize the image path?

As mentioned by the others, this is simply a matter of copying the backup photos to the original locations, and making sure the disk has the same name/letter.

I point out for clarity that you are not transferring the image INTO Lightroom, this is all done using your operating system. Importing the photos into Lightroom is NOT a step in this process. If you are planning to import the backups, don't.

If the photos are placed in the same structure as the import and the disk is wrapped with the same name or drive letter, then there are no problems.

Otherwise, you have to tell LR where it finds the photos.

https://www.lightroomqueen.com/lightroom-photos-missing-fix/​

Adobe Lightroom - Find moved or missing files and folders 

If your backup drive file/folder structure EXACTLY matches the failed drive structure it will be easy but if the photos are backed up in a different structure than it will be very difficult if not almost impossible without extensive work.  Which scenario is your backup drive?