You did everything right in terms of troubleshooting. It sounds like the LCD has gone bad. Luckily, this is is a relatively easy fix if you take a look at the [guide|22872|iFixit guide]. This will also provide the link to the kit you will need in order to do the repair if you do not have the tools.
Just so you are aware, the part does not include the home button or the front camera assembly. These will need to be transferred to the new screen. Be sure to use heat to remove the button and camera because simply pulling them off can tear the cables.
[guide|23102|Home Button Guide]
[guide|33519|Front Camera Guide]