LCD, but I'd advise that you save yourself some money and pay a repair shop to do it.
There are too many posts here along the lines of "I replaced my screen,now xxx doesn't work. How can I fix it?" Working on a cellphone isn't like cutting firewood. One tny slip and it can cost almost as much as a new phone to fix the damage.
I owned a repair shop. I hired people wth experience in repairing ,icroelectronic devices. And I still had them practice on parted phones (phones so bad that we just kept them to steal parts from) to learn things like replacing screens. These phones aren't like an old Nokia, where te screen just drops in. You have to take things apart - without breaking anything. That means knowing that, n this phone you can lift the screen up 3 inches to unplug the cable, but in that phone you have only 1/8" play, and you unplug the cable with very thin curved needle-nosed pliers, or you'll rip the connector off the motherboard and have to replace the whole board.
If you insist, it's the screen (the LCD, including the backlight), but I really advise against DIY in this case.
(I'd also advise cooling the relationship off a bit until after she successfully completes an anger management course. This time it was a wall. Next time it could be something more important to you.)