The old-style way to get a perfect glossy mirror shine on wood is by using French Polish which consists of a technique that is rubbing layers of shellac on the wood. The process is very time consuming, and due to the stickiness of the shellac, it can be hard to work with.
There is a less demanding method that you can do which is to rub-down the hardened finish, which will be either lacquer or varnish. Lacquer is more of a non-curing finish that is more like the shellac rather than the varnish, and due to the fact it dries faster, it means you can layer more easily with varnish. This will give you the glassy smooth finish that you are looking for.
What do I do?
First of all, start with sanding the bare wood to remove the old finish and keep it smooth. Finish the sanding process by hand sanding using an aluminium oxide sandpaper that is 150 grit. Ensure that you sand the wood with the grain and not against it. Use fresh and dry cloth to wipe away the sanding dust.
Now you want to fill the grain with either a water-based or oil-based grain filler, suitable for wood. This step is essential if the wood has a more open grain like oak, but you can achieve the mirror finish on all the varieties.
Apply the filler with a paintbrush, and then you can scrape it flat with a paint scraper then leave it to dry over several hours. You can sand it flat using your hand with 220-grit sandpaper ensuring that you work with the grain.
You can now spray one coat of your lacquer or apply varnish with a spray or brush whichever you prefer to do. You need to wait until it no longer feels tacky. Lacquer will take around thirty minutes while the varnish will take a few hours to dry. Once it no longer feels tacky, you can sand it down using different sandpaper that is 320-grit and again wipes off the excess dust. Now you can apply one more coat of the lacquer or varnish. If you are using lacquer and spraying it, then you can add another two coats ensuring that you are sanding the first one before you apply the other.