looking for some k/jdramas where a family member all of a sudden has to take care of a kid  or teen 

Usagi Drop - it is movie but should be good (I have not seen yet). :) There is also anime version.

  • Stepmother and daughter blues
  • My girl (2009)
  • Zenkai Girl
  • Stepfather Step
  • Oh my geumbi
  • Oh my lady
  • Stars falling from the sky
  • Mama: Nothing to fear
  • Super Daddy Yeol
  • Movie:
    • Bunny Drop
    • Baby and I
  • Non korean/Japanese Drama
    • Old Boy
    • Down with Love
 kardasx:

Usagi Drop - it is movie but should be good (I have not seen yet). :) There is also anime version.

i been wanting to watch this but cant find a spot to watch it

im sure you will pretty like to watch "baby and me" its full of taste u need

Come! Jang Bo Ri - Makjang family drama, FL takes care of the villain's daughter who she abandoned. 

This is an oldie but a goodie: Stars Falling from the Sky

If you're looking for a comedy with this concept 

Eulachacha Waikiki

18 again has Wi Ha Joon's char taking care of a lil girl, but he is not the lead

  • Boys over Flowers Geum Jan Di needs to take care of her little brother, while their parents have to work elsewhere for income. They live apart for a while.
  • Doctors Yoon Hye Jung (teenager) has to go live with her grandmother. (Her grandmother dies and there is a time skip when she becomes a doctor).
  • Jealousy Incarnate Pyo Na Ri lives with her younger brother (teenager), their parents passed away.

Movie:

Scandal Makers    

Awesome movie with Park Bo Young.

Rough synopsis:

She appears in front of her father, with her own child in tow. An unsuspecting and popular radio host bachelor and womanizer becomes not only a parent overnight but also a grandfather. So he has to take care of his now kid and grandkid!

So this fits your description perfectly, and then some.

Also, this is a very traumatic drama:

Mother    

Synopsis: A teacher witnesses one of her student (girl) being abused by her stepfather and mother. After careful thinking she decides to flee (or kidnap) the girl, and they become mother and daughter as they flee.

Oh! My Girl!! - jdrama - poor uncle suddenly have to take care of his rich niece.

The Last Empress    

In a way, but it's not the main theme, just a side theme.