I put my U5i to charge and when I woke up in the morning it was shut down and I can't turn it on...In two words the phone died.

Did you drop the phone in water recently? Did you get it wet in the rain? Apart from the likely possibility that the phone is actually damaged/battery damaged, this happens if the power button gets damaged. When you put the phone to charge, in the middle of the night the power button "acted" as if it was pressed and held and the phone turned off. Now when you press the power button, nothing happens because the power button is damaged.

Take it to a repair shop and check the power button for damage. It is replaceable and should not cost a lot.

Does your phone have a removable battery? If yes then you might be in luck. If not, then just take the phone to repair shop and get the power button checked and repaired.

If battery is removable I know that it is possible to turn on the phone this way:
1) Plug in the charger to the phone
2) Remove the battery and insert it immediately. Do this multiple times and the phone should turn on without having to press the power button.
Hope that works.