How can I make the detail of this map stand out better?
I would call inlaying metal wires into slate a high-risk project in general.


You already managed to carve the groves - congratulations for that!
Slate has the tendency to chip or break when hammered
Colored wire for jewelry often has a very thin coating of color over a soft metal core (often aluminum). Hammering that into the groves could damage or remove the color coat.


Add to that the fact that the contrast between slate and soldering wire is not very high - both appear silver or grey, depending on the lighting.

My first advice for this project is to use a contrasting color for the groves. Anything but black, white, silver or grey will stand out better.

Because of the high risk of breaking a sheet, I would not hammer anything into the grooves. Filling the grooves with a liquid promises more success. 

One idea that you can test for a low price is good old Window Color. Buy a bottle of Window Color contour paint - these have a thicker consistency and a higher pigment content. Test the paint on the back of a sheet and see if you can peel it off in case something goes wrong. If you're satisfied with the test, squirt the paint into the grooves and let it dry. You can thin the paint by adding a few drops of water to the bottle.

A better shine might be possible by using metallic nail polish. I suggest transfering the nail polish into a squirt bottle with a thin nozzle that fits into the grooves. Both (bottle and extra thin nozzle) should be available online or in good crafting stores.

Metallic permanent markers might work as well, if the tip is thin enough to fit into a groove. Both nail polish and permanent marker can be removed with nail polish remover, but the grooves probably make it harder to get all of the paint out.

Another alternative that promises an even and shiny result is enamel color. These are often used for model painting and have a high pigment content, like this example by Revell. The only disadvantage is that I don't know how you could remove it in case something goes wrong, but I'm sure the manufacturer will provide more information. Use this paint with a squirt bottle as well.

In any case, my experience with slate as a crafting material is this:


Before applying any paint, wipe the slate off with nail polish remover to remove any oily residue
Slate needs extremely long to dry. It took almost a whole day in my case and in the meantime the slate looked blotchy. It's only dry when the whole sheet is a uniform light grey.
Test any paint or nail polish on the back side. The structure of slate can make some colors run and bleed into adjacent areas.