Reparing a water damaged drywall & doorframe
If you have never done anything like this you might want to get a pro to do the repair. If the wall was wet long enough from behind to rot a door jamb that you see, the framing behind the jamb and drywall will most likely need repair too, possibly the bottom plate replaced if pressure treated material was not used.
You can do some of the demo work to verify what you really have, but the door jamb should be replaced. The drywall needs to be cut out beyond your "poke holes" at least 1 ft. beyond each side and about 16" up from the floor to start. If there is mold growing on the back side of the wallboard, remove more until it is gone. Once the drywall is removed and cleaned up, next would be evaluate the extent of the rot. If the wall is non bearing it will be simple, If it is a bearing wall, it will take a pro to determine what needs to be done. A note on drywall repair, whether the repair is 16" tall or 2" tall, or 4' wide or 8" wide, it takes the same effort to repair and it will still take a full sheet of drywall, drywall tape and mud, so don't be squeamish on what needs to go.

You may be able to salvage the skirt boards if they are not rotted from the back side. If they have mold, it can be cleaned off and the back can be painted with primer that has mold inhibitor in it.

If you can find a reputable, licensed handyman with references and pictures will help. A picture from far away that shows the whole project will not show what goes into the job, knowledge and quality wise. Get references from the handyman so you can call the people he worked for that he did the job for will really tell you more than the pictures. The pictures just show he did it, the references tell you how well he did it.