You can't use the factory recovery like that. If you move or copy the factory recovery partition on it's own to another hard drive it won't be usable anyway. In any case, if the existing hard drive is so bad that cloning doesn't work, copying from it won't work either.
You should have made a set of recovery discs when your Dell was working okay, or created a disk image of the entire HDD and saved it to your external one.
As it is now, you will have to replace the internal HDD with a new one, then contact Dell and ask them if they can send you a set of recovery discs for that particular model.
A little bit of attention to creating a system backup earlier would have avoided your current dilemma.