Why Does The Olympics Only Include Taekwondo As The Only Eastern Martial Art?

To understand why the Olympics has included taekwondo as a sport and no other Eastern forms of martial arts you will need to understand the standard by which they choose which sport to include and which they will not. There are 2 issues that stand in the way more than anything for the inclusion of other martial arts.

Taekwondo | International popularity

First, the sport must have enough interest internationally. For it to be included in the Olympics it must be a sport that is practiced all over the world and not just a single area. The standard is currently set that a sport for men in the summertime Olympics must have populations of people in seventy five countries over the span of four continents. For women sports it is reduced to only needing practitioners in forty countries over the span of only three continents.
This rule alone is enough to prevent most styles from being qualified. Most styles are practiced in only one country and for this reason cannot be considered. Many are interested in seeing styles such as Brazilian jujitsu and Muay Thai included. These sports are popular in both their country of origin as well as in America, but mostly are not so elsewhere. It is possible however, that as these martial arts become more popular they could be included later.

Taekwondo sports must fit with-in the stated ideals of the Olympic Games

The next stipulation that must be met before it can be considered for inclusion is if it has widely accepted championships on the worldwide level. It also requires that there be a governing body that is considered international. It is not enough for someone to simple establish a few schools in two or three countries and his own International headquarters. This would not be considered official and therefore would not meet the requirements.
In addition to the above criteria needing to be met, the sport must fit with-in the stated ideals of the Olympic Games. If a martial art is designed for warfare or street protection, then it does not meet the criteria. So arts like Krav maga, would never be included as it is specifically designed for street warfare against terrorist.

Taekwondo is specifically designed for battle and street warfare

It is also necessary that those involved in the sport have enough drive to see it put into the Olympics. Japan had a deep and driving desire to see Judo included and its practitioners were equally motivated. Taekwondo was created in part to renew Korean national sentiment and for this reason all of South Korea wanted to see this sport included. It had also become popular with practitioners all over the world which shared the desire to see its inclusion.
Many martial arts are specifically designed for battle and street warfare and the very nature of their design is simply too lethal for a sporting event. Judo and Taekwondo were specifically designed for sporting events. While they take from many war time practices, they were specifically developed for the ring.

After all of these other considerations are the fact that some feel that Judo for instance, has lost its way since being added to the Olympic Games. Some feel that since that time, they focus on strategies of strength and speed. The original focus of this sport was on getting the greatest effect while using a minimum effort. This has caused the sport to be watered down and removed its origins of a true martial art, in their opinion.

So you should understand that there are many factors that remove the possibility of many of the martial arts from being included. First and foremost is that many are indeed meant for warfare and street protection. In addition, many simply do not have the organized establishments in place that are required for inclusion.