Just a standard 2-prong G4 bulb. Any number of replacement LEDs will work as long as they have the two prongs. If you look at the Winne Parts catalog for your light you'll see a tiny picture of the bulb right above the listing.
Now a couple caveats (having replaced all mine...) you most likely want downward facing discs. Check the size, as you have limited space. On mine (the brass version of yours) there was a ceramic heat sink in the fixture to handle the heat of the Halogen bulb. I had to remove mine (simple screwdriver work) to make room for the LED disk light.
If your lights are on dimmers, make sure you get LEDs that work with them... not all do. I don't have a dimmer switch so wasn't an issue for me.
For what it's worth, I used these http://www.ledlightsworld.com/dc12v-...aaea3522cdab1f
but there are many available in today's market.
As you look and compare, think about how bright you need. I tried ones with 10 LED on each disk, which I decided were too bright, so I dropped back to the ones above that have 6 LEDs. Also, you'll have to decide on the color "temp" of the LEDs. I went with Warm White because I didn't like the strong pure white some offer.