If you want it good, you will need to redo it. Good thing is - you don't need to worry about destroying surface too much. With chemical damage like that, surface is effectively destroyed beyond repair already.
If you got a veneer over wood, and furniture is made in an old way, chances are it's glued using bone or skin glue. Heating it up to 40 or 50 degree Celsius will make glue unbound, and you will be able to remove veneer easily. If this does not work, heat might damage surface, but oh well, not a problem now.
If it's not hot-debounding glue, you need to sand it. Oscillation electrical sander would be my bet - less chance to make waves than if you do it by hand, and less chance to sand too much than if you would use belt sander. In my country hand-held belt sanders are often nicknamed tanks not only for their look, but also or their delicacy and precision (or lack of). If you are going to do it by hand, and don't have practice, read about sanding first.
Now, there can be three results:
- You sanded thorough paint only
- You sanded thorough veneer as well
- You were lucky and only sanded thorough damage
If 2, I would highly recommend putting some veneer there. It will be painted so it might be cheap all right, but the only reason to put veneer under paint is to hide construction connections, not to look good on it's own. On the other hand, it is the top - the only part that's ofthen different by design. You may put dark/black decorative veneer, lacquer it and call the job finished. It may look better than original that way!
If you don't see that you sanded through veneer, you don't need to care if it's really there at all. You got layer of wood so you are good to go. Put thin coat of paint. If you don't have original paint now, wait for first layer to dry and put another one. if old paint is there, one might be enough. on the other hand, sanding to the naked wood helps avoid compatibility problems and weak bonding between different paints.
If black isn't enough, and you want finish like "piano black", you need to polish your black paint, and then cover it with many thin coats of compatible high gloss transparent lacquer. If feasible, buy them at the same time, making sure assistant in shop knows you want to use them together. I think that details on piano black would make this answer even longer, and it's material for another question. I can't even tell if original was piano black or regular black.