What should I do about this unfinished corner?
This will be a variation of what may have been mentioned already, but this is how I would handle it. 

For the tall vertical piece, assemble 2 pieces of 1X material to make an outside corner that will mimic the left side wall where it meets the door. At the top add an additional piece of 1X to fill the void at the wood joist or beam at the top of the door. The idea is to bring the new material out in the same plane so when it is all painted out, it all looks like the same wall around the door. The only difference being new surfaces next to the older surfaces. The illustration will help depict what I mean, take note of the stunning green parts of the drawing that is referred to