How long I would have to be in Korea before getting approved for command sponsorship to bring my family over?

It's two years accompanied. The source document is the Joint Federal Travel Regulation Appendix Q. It lists all OCONUS tour lengths to all locations for accompanied and unaccompanied tours.

The way your question is worded, sounds like you want to come to Korea, and THEN get command sponsored. A tour is 2 years mandatory if with family, but you should get command sponsorship on your orders BEFORE you come to Korea, and bring your family when you PCS. It's possible to get command sponsored after you arrive, but it might be difficult. You could be assigned to a unit that may not allow for sponsorship, and you have no option. If you have sponsorship in your orders, they will place you and your family somewhere that will accommodate. Do it on the front end, or you might find yourself in Korea for a year... without your family.

As of right now a command sponsor tour in Korea is 2 years. If by chance your family departure get delayed, your time will still be 2yrs because it’s the Soldier who has to do the 2yrs, not the family. Good luck to you.