Know Before You Go
While it is possible to fly into Leh and simply drive up 25 miles with any local taxi, this is strongly discouraged. The elevation at Leh itself is 11,483 feet (about 2.2 miles), which can cause even healthy people to suffer effects of AMS, possibly even fatal, and traveling to the top of Kardung La adds an additional mile of height.
Most people taking up the Kardung La challenge begin at least several hundred miles away at Manali (an elevation of only 1 mile) and space out several overnight stays at or below 2 miles high. This takes several days but the trade-off is that at each rugged pass climbed the scenery changes in incredible ways. In addition to breathtaking views (literally), this journey will span a few cultures as well. Be respectful and carry lots of water, and possibly extra fuel. Permits must be obtained along the way and may be limited, so be prepared to stay overnight with only a few hours of electricity.