Hello lalago87 and welcome to Seven Forums.
I think I understand what happened and what you're trying to do. But just for clarification, are you trying to do a system restore or a factory restore? They are two very different things. I'm guessing it's a factory restore because you get a screen saying the computer is being prepared for first use. So if you're really doing a factory restore, are you accessing the hidden recovery partition on the hard drive or are you using recovery disks? If recovery disks, they may not have burned correctly. You should be able to use the Dell Support site for specific instructions on how to access the hidden recovery partition. How To Restore or Reinstall Microsoft Windows on a Dell Computer | Dell
According to the listed specs, your sister's computer is running 7 Home Premium and you have a 7 Pro disk. If you're trying to repair the computer, you have to use a disk that is the same version that is installed on the computer. Home Premium to repair Home Premium, Pro to repair Pro, etc. Not only that, they have to be the same architecture. In other words, if the installed OS is Windows 7 Home Premium 32-bit, then the disk being used for the repair must also be Home Premium 32-bit. Repair Install
At this point you may want to take the opportunity to do a clean reinstall and get rid of any factory installed bloatware. One of the Forum experts prepared a tutorial on how to get a perfect clean reinstall. And if you need a free, legal Home Premium 32-bit installation disk (because that's what the computer's Product Key is tied to) then Step 1 explains how to do that as well. Clean Reinstall - Factory OEM Windows 7