With the help of Google, a bright man named @jbrennan51 gave some tips in another ifixit forum.
Here are a few more ideas for you to try.
Restore iPhone in DFU Mode - Try to restore your iPhone in DFU mode in order to fix iTunes error 4005. Here are the steps to enter in DFU mode:
Connect your device to the Computer.
Turn off your device.
Hold Power for 3 seconds
keep holding Home and hold Power for 10 seconds.
Now release Power, but keep holding Home.
iTunes will detect your iPhone in Recovery mode. Now you can start the restore process.
Replace your Battery - In some cases, replacing the battery can fix the iTunes error 4005. The main problem is because the battery have a low power. We recommend you to replace with a original battery.
Replace the Dock connector - If you used some non-original chargers, your dock connector can be damaged and have a bad communication with the usb cable. By replacing the connector, the communication will be perfect and you will fix the error 4005. Make sure you are using a original certified cable when you restoring your device via iTunes.
Restore Without a LCD - Disconnect the display from your device completely, connect to iTunes and try to restore it. If you have hardware problem in the display, this will bypass the problem. If you still get a Error 4005, then move to the next method.
Replace Proximity Sensor Flex Cable - Try to replace the Replace Proximity Sensor flex to fix Error 4005. If you still get a error 4005, try to replace the front camera flex cable also.
Remove compass IC - The compass chip can be the main problem you getting iTunes error 4005. Try to remove the Compass iC to get rid of this issue. We must to note that you need to have a technical skills to do this method.
Conclusion: Usually, these methods would totally heal the Error 4005 issue on iTunes. However, in case the problem still persists it’s suggested that the Apple device is taken to an Apple service center immediately. Don't forget to share this article and post a comment below.
Here is the forum link to where this was found: stuck in dfu with itunes error 4005
There are also some more helpful ideas within that forum.