Any type of dancing has been proven to increase flexibility and coordination. However, ballroom dancing (couple dancing) works your cognitive ability to a higher level by making both the left and right hemisphere of your brain work at the same time. Your brain needs to learn to steps of each figure, your brain needs to put all the steps together in some sort of routine, your brain needs to tell all your muscles how to walk into those figures and how to lead the lady, and finally, in any social dancing situation, your brain is required to make immediate fast decision to change direction or dance routine to avoid bumping into another couple.
Other types of solo dancing does not make your brain work on the last part of the above description. You learn a piece, you put the music on and you dance that piece of choreography. In couple dancing whether in social or competitive environment, your brain is constantly evaluating the situation to decide if 1)continue with the pre-determined routine or 2) change it to avoid another couple.
As for time and money, if you are serious about learning couple dancing, I believe that it is important to make time to learn this activity. If you already don't have time because your time is filled up by martial arts, maybe you can cut down 1 day of martial arts and learn dancing instead. The money that you spend on that day of martial arts will be put towards the dance class instead. I'm not sure how much martial arts lessons cost but dancing is quite expensive, even group classes.
Staying in shape: social dancing gives you a peaceful workout. You stay in movement but not much cardio work. In the beginning, your arms might be painful as you might not be used to holding your arms in certain position for long periods of time (standard dances). However, if you are looking for staying in shape and having a good workout, competitive dancing is where I will point you to. You train at the gym and on the dance floor. Your competition day will provide you with both aerobic and anaerobic exercises!