Replacing the screen at this point is like doing heart surgery when all you know is that the patient is unconscious.
If the phone doesn't respond to a power-off attempt (power button does nothing), then the touchscreen is not the immediate problem. If the iPhone's system has crashed and needs to be reinstalled, or a component failed on the main circuit board, then even a perfect touchscreen won't work.
The same goes for the reset attempt (power and home buttons held in until Apple icon appears on screen), or a DFU attempt. None of these require a working touchscreen, so if those didn't help, then the touchscreen isn't the likely problem.
Before you tried to use DFU mode (usually, the "if all else fails" approach), did you try a regular Restore? Did you try to put the phone in Recovery mode?
What happened? (Or, what didn't happen?) Was the iPhone recognized by iTunes, or not?
Also, consider that there could also be a problem with iTunes software, or the sync cable, or any other number of things in addition to the phone. Can you successfully sync other iPhones/iPads/iPods to that computer, using the same cable? Have you tried restoring that phone using a different computer?
