There are only 3 ways to transfer data to new phone:
- iCloud
- Quick Transfer
- iTunes
Since you can rule out iCloud and Quick Transfer, that only leaves the iTunes option. This is only available if you have plugged your phone into a computer and backed it up there. Since the phone will not turn on, I doubt iTunes would recognize it now, but if you have a computer I would try to plug it in and see what happens. It would depend on what is wrong with the old phone. If the display is broken and just shows a black screen, it may still be possible to backup to iTunes.
You are probably going to have to start fresh and loose your data. It comes down to how much the old data is worth as you would have to pay to have the old phone repaired to be able to transfer the data without a backup.