I've seen plenty of dramas where the leads either marry in the very beginning or the very end of a drama, but is there any dramas where they marry in the middle? 

Well, not in the middle but maybe early on:

Fated to Love You    (closer to beginning than middle though, but still some episodes in)

Eulachacha Waikiki    (fake marriage near middle)

Full House     (contract marriage, probably closer to beginning than middle)

The Last Empress    (probably after 20% way in)

Dong Yi    (becomes concubine in middle)

Empress Ki    (becomes concubine in middle)

The Story of Minglan happens almost exactly halfway.

A lot of long dramas had the main or secondary couples married in the middle of the drama or a few episodes before the end of the show, these are two of the ones I loved.

Golden Bride (the main couple marry in the start but since it's an arrange marriage their real married life doesn't start until like 20+ eps out of 64 after they fell in love)

Story Of Ming Lan (they get married exactly in the middle of the story)