You have a project ahead of you. One, you don't want to have a repair guy come in and fix some cosmetic repairs. We're assuming the table is set, level, and the slate seams are set with beaswax, bondo, or what have you. If this is not the case, start over, with a table mechanic, and do it right.
If you want to do it yourself, not my recommendation, but find a guy with a strong back. Gently, very gently, lift the frame with the strong guy's back, then quickly slide a piece of cardboard under the leg bottoms. You only showed one. I'm assume all look like this? Or is it only one?
The rest us up to you. Brasso, ammonia and water, with a scrub brush, either will do the job to get the gunk off. After a lot of drying, and I think wet sanding with fine sandpaper, then I would paint with black enamel. Tape the extremities. It's up to you at this point, but leave the cardboard underneath until everything has dried.
Then, get that same strong dude with a strong back to slightly lift by the frame, not the rails, to get each piece of cardboard out from underneath, assuming you have a cleaned, and painted surface.
Then, roll some balls slowly down the length of the table, and back and forth on the sides of the table. If it rolls off, and it probably will, get the same strong dude to get down and gently lift by the frame, and turn the feet where they need to go. Clockwise will raise at that leg, counter-clockwise will lower. If that's hard to remember, right to raise, left to lower. If adjustment is needed, do a careful half turn on each side you are trying to adjust. Don't adjust one side, or length without adjusting the other, assuming they starting hitting the floor with equal pressure.
Somehow, I have a feeling there is more to this story than meets the eye.
But, as always,
All the best,
WW