How about magnet wire? Since you will only power a couple of LEDs you can use a very fine gauge wire. You could attach (glue?) the wire to the inside of your support legs and mount your power supply under your table.
I like Danr's idea about Magnet Wire which is a different animal than Kynar. For the sizes in question for LED wiring in your structure, the enamel coating/insulation on magnet wire only adds about 1 mil (0.001") to the diameter of the underlying wire. Typical Kynar insulation adds maybe 10 times that thickness.
So, for example, in photo above the underlying #34 magnet wire diameter is about 7 mils and the coating adds about 1 mil so that's only 8 mils total diameter. The #30 Kynar is about 10 mils underlying wire plus another 10 mils of Kynar insulation so total is about 20 mils. Obviously your #28 Kynar will be a bit thicker...say 25 mils overall.
The reddish-brown coating for the magnet wire is very common and might blend in without painting because it's so thin and hence probably not noticeable. Magnet wire can be tricky to work with since you must remove the coating - either by meticulous/careful scraping using a hobby knife or preferably by "burning" it off with a soldering iron. Of course the biggest practical issue is where to get magnet wire in small lengths (a few feet in your case). As shown it typically comes in spools of thousands of feet!
If you end up wanting to try magnet wire if for no other reason than to learn about the stuff for a future finescale modeling project I will spring for a postage stamp and send you some.