Backlight filter blown after screen replacement

If all that's busted is the backlight filter/circuit, you don't need backlight to get data. As long as the phone powers on, gets image and accepts touch you're usually in business (also need either WiFi or USB to get data). Power up the phone and shine a flashlight at it. Is there any image on screen? If no, this is bigger than just the backlight, you've also got no image. In which case, that's likely to be an issue.

As a note for sake of clarity, the electrical systems which produce image and backlight are different. Which is why your phone can light up, even if there's nothing on screen.

If you cannot see what's on screen to type in a passcode, generally speaking, you can't get data. If I recall correctly, there are forensic tools for some older devices to enable you do enter it with a keyboard rather than on screen. But even that requires a certain type of jailbreak. It's not something the every person who have access to.

Cross my fingers this is just a backlight deal and you'll be all set.