To DIY, you'll need to remove the flaking paint and featheredge the rest that's solid, where it meets the metal. Use 400 grit, then 600 to 800/1000 until you can't "feel" any transition between the metal and the existing paint. If you can feel the edge between paint and metal, you will see the edge when all done.
Clean the metal with naval jelly ( or acid etch it) then use a "self-etching" primer on the metal, then scuff the self etching primer with a red Scotchbrite pad prior to painting. Degrease well too, prior to primer and paint.
If you have a seam/body panel line somewhere near the repair, you can tape off to there and respray base/clear on the entire part (this looks best if you can do it). If the part segues into a larger part of the door, then you'll have to try and blend it. It won't be seamless, but it'll be better than your current situation.Some good wetsanding and buffing afterward can make it less obvious.
The shaker cans also won't be as glossy as the oem paint, and will take months to fully cure, since they're uncatalyzed, so don't wax it for a while.
Yeah, you can just use brush on touch up paint, but it won't look too good. Do that only if you want to get paint on the bare metal (you still will need to use the primer) and buy yourself time until you get it to a body shop.
It's a lot of work really, so if it sounds harder than you thought, maybe look into a pro body shop. They wouldn't charge more than $125-$200 for something like that, generally speaking. Good luck.
