If you can afford to lose the data on it, you might try using 'Pwned' DFU mode, which is a sort of factory firmware flashing mode similar to recovery mode, or Android's Download Mode. Once the device is in DFU mode (steps below) iTunes should tell you that it has detected an iPhone in restore mode. Choose to "Update & Restore", and iTunes will download, verify, and install the newest iPhone 5 firmware to the phone. When it is done the device should be accessible. If the screen still does not respond afterware but iTunes continues to interface with it, it is an issue with how the screen was installed, in which case you should (obviously only if you were the one to replace it or know what you are doing) open the iPhone, find the ribbon cable connected to the screen, disconnect it, ensure that there is nothing that would cause interference with the connection (dust particles, fuzz, etc.) and reconnect the ribbon cable. Apple uses a strange ribbon cable design and I have found in the past that it sometimes seems as though one is fully connected when it is not.
Steps to enter DFU mode:
- Disconnect the iPhone from your Mac.
- Hold BOTH Power & Home for 20 seconds.
- On the 20th second, let go of the Power button but KEEP holding the Home button for an additional 30 seconds.
- Your screen should be black, reconnect your iPhone to your Mac and if it worked iTunes should report that the connected iPhone is in recovery mode and needs to be restored.
- Either Click "Update & Restore" to auto-download and install the newest firmware for your device from the Apple servers, or hold down the "Shift" key while clicking the button to install a custom
.ipsw firmware file (will only work with firmware built and signed by a licensed Apple Developer account).