any decent enamel will work. Make sure its the "right" black though. try testing out little drops in one of the body cavities to make sure its not too flat or too shiny.
Then you can do whats called the "drop fill" technique. basically get a tooth pick, get it to where a drop of paint is gooped up on the end, then "drop fill" it in
. try to keep it flat. then do the same with the nail polish, but with the clear, you want it raised so you can sand it to meet the surface of the clear coating. Nail polish clear is actually really really good because it doesnt fade and dries EXTREMELY hard. Before you do any painting, make sure you feather the edges of the clear coat 1) to promote proper adhesion and 2) to prevent the existing clear from chipping away still.
Best way to do the sanding is to mask off the areas around the chips so you dont do areas you dont need to. Get some 600/800/1000/1500/2000 grit sandpaper, and wet sand the area. I see your guitar has a lot of swirls and scratches, you may actually want to do the whole guitar. After you do the 2000 grit, you can use some light rubbing compound to buff it out, then use some polishing compound. You can "hand rub it" but prepared to do a lot of work, even for a small area. If you dont have a good buffer, go get one of those "headlight restoration" kits with the drill bit attachment (20 bucks). comes with everything you would need actually. Dont use the sandpaper, because that is overkill, but it has a very decent buffing wheel attachment, and has rubbing and polishing compound that comes with it.
I do a lot of paint restoration on cars (basically any time my wife goes through a drive through car wash -.-), and the system is the same for guitars. I'm debating on doing a complete, step by step, walkthrough on how to do the clear coat sanding and polishing.
worse coming to absolute worse, go get a black permanent marker, and be REALLY careful with the nail polish, and you may get away with a 5 minute fix.