Traditional styles offer complete mind/body conditioning and for your goals I would choose one of them.
The one you choose will of course depend upon availability but you should also consider what you bring to the table:
Don't go crazy with this martial art people, these are generalizations!
Karate styles emphasize one hit power
kung fu styles emphasize speed and technical skill (but with more than 300 styles there are exceptions)
Tae Kwon do/tang soo do emphasize kicking
judo, aikido, tai chi, jiujitsu emphasize skill and using the opponent's force against them.
For flexibility no other style is more demanding than monkey style kung fu. Next would probably be eagle claw kung fu.
A qualified instructor is very important as well as what you bring to the table. The styles that require extreme flexibility are best started early in life because like ballet, beginning young allows the body to adapt when it is still pliable.
Your demeanor plays a role too, Shotokan karate for example, is very "in your face", which is great for aggressive personalities and those who are physically strong but not as well suited to smaller people (in general).
For application, there is no better art on the planet than tai chi, provided the instructor has learned it as a martial art but many, maybe even most instructors in the U.S. learned it as a health exercise only. That being said, it also takes longer than any other art to become even mildly proficient plus, while you gain some flexibility, it won't lead to doing the splits or anything like that unless you practice that on your own.
You have to be careful regarding instructors because, depending on your location, you will find all kinds of "masters" (a title thrown around today like "black belt" was 30 years ago.
To understand what I mean consider this: in Karate, traditionally the practioner was given a white belt. Over time, through use (dirt), the belt would eventually turn black. When this happened, they were considered a "first grader" or "black belt" (so figure at least ten years) but they were not considered expert until their belt again turned white (from fraying) another 10 years? maybe or more! Today, schools actually advertise GUARANTEEING a black belt in 2 years or less!
Of course, being America, there are teachers who claim to teach slowly because they are "serious", "traditional" or whatever but in reality do so because they don't want to run out of things to teach you!
Once you decide on style (google, youtube to see differences) find some online forums for that style and ask about the proposed instructor/schools there to get first hand knowledge of the instruction.