It sounds like someone tried to replace the FPC connector before. The plastic of the connector and the epoxy on components on the board around the FPC connector will blacken and turn sooty if overheated, eg. with a hot air station. Replacing the FPC and LCD may be required to fix it, but you may have other issues as well. The most likely one that would be caused by a short is a dead backlight circuit - very, very, very common on Mini 1s and 2s. That would cause it to seem like the screen is dead but careful examination in the dark would find that the LCD is on but extremely dim (no backlight). So check that first just in case the FPC isn't the problem. Repairing that is much easier than the FPC replacement.
FPC replacement really requires some sort of optical magnification to ensure the pins are lined up properly, and then to be able to check that each pin is soldered to the pad. The procedure is basically:
1. Disconnect the battery! If the backlight circuit isn't damaged already it will be with the slightest wrong move. Disconnecting the battery prevents this.
2. Add generous amounts of a good electronics flux around the old FPC connector, but try to keep it away from other components to avoid lifting them.
3. Use aluminum or copper foil tape as a heat reflector on other components, exposing only the FPC. You can find aluminum foil tape at any hardware store that sells metal ducting.
4. Heat with a hot air gun set at around 260-300C while lifting up the FPC connector. Use a small nozzle and moderate airflow to avoid overheating the board or desoldering other components. Use sharp tweezers or an X-acto type knife to lift the FPC.
5. Clean the pads until they're spotless using new flux (regularly cleaned off with isopropyl alcohol) and desoldering braid.
6. This is the tricky part. You need the new FPC to sit flush with the board so that all pins will make contact, but you also need solder on the pads obviously. There are 2 ways of doing this. The "correct" way (I say "correct" because it's how it's done at factory) is to use solder paste applied carefully in small amounts on the pads. The suspended solder balls in the paste will allow the FPC to sit pretty tight to the board when placed. Then you heat and once the paste hits the melting point surface tension will pull the solder to the pins and the pads. The other option is to carefully apply as-close-to-equal-as-possible amounts of solder to each pad, verifying that the quantities are the same under a microscope or similar. You don't need much. Once it's looking good apply a very thin amount of flux, align the FPC, and heat the solder while using tweezers or similar to keep a slight amount downward pressure on the FPC. Once the solder melts the FPC pins will drop down to the PCB.