OK folks, this is the solution to the problem of iPhone/iPad stuck on Apple logo screen; Cannot shut device down; iTunes and computer will not recognize device after plugging in to USB port: (At least it was the solution for me. Worked for both iPhone 5c and iPad 2 Retina.)
PLEASE NOTE: Restoring your device using iTunes may result in the resetting of your device to factory condition. Data may be lost. If you have previously backed up your device to iTunes, you can use this backup point plus iCloud to restore most of your data and settings.
Plug device into computer running iTunes, using the USB cable. (If iTunes recognizes your device, you can stop here and restore your device - either as new or from a previous backup)
Reset your device - while plugged in - by holding down the home and sleep buttons for several seconds. The apple logo should blink off and your computer and iTunes will recognize that your device is plugged in.
Proceed with restoring your device using iTunes on your computer. If iTunes doesn't automatically recognize that your device is in recovery mode, click the name of your device in the left pane of the iTunes window, then proceed with the restoration of your device. If you have ever backed your device up to iTunes, you can choose to restore from that backup point. Otherwise, you will have to restore the device as new (factory reset) and will lose your data - however, all apps, free and paid, can be redownloaded from iTunes. Also, if you have backed up any data or media from your device to iCloud, you can restore that data using your device and a WiFi connection.
Hope this helps.