Turns are Spring, Summer, Autumn. Start with 1231.  Keep a note of the year & season.
All armies retire to quarters in the winter - for the Koreans, anywhere and for the Mongols, the oval at the north.

The Mongols start in the oval at the north with a 12 element army.
The Korean army - also 12 elements - is anywhere.
Each season first the Mongols then the Koreans move.
The Korean army can move 2 areas but must stop when an enemy army or town is met.  The Mongols can move an unlimited number of areas but also must stop when an enemy army or town is met.
When an army reaches an enemy town it besieges it. If there is an enemy army there a DBA battle is fought first.
In the turn an army first besieges a city it must roll 4,5,6 to take it.  On the next 3,4,5,6, and the next the same.  No longer sieges are possible as winter intervenes.  Note that Kuju needs a roll of one higher when Mongols try to take it (5,6 then 4,5,6) and Ganghwa three higher (7, which is impossible, then 6). Each failed roll causes the besiegers to lose 1 element.  There is a 1 in 12 chance the general is killed (roll 6 followed by a 4,5,6).

If Ganghwa island falls the Koreans submit and the game is over.

At the end of each Autumn roll for Korean submission. The Mongols total up 1 point for each city or area they currently hold and roll a D10.  If the score is equal to or higher than the points held Korea sues for peace.  The Mongols withdraw but retain all captured areas.
After Autumn also roll for the death of the khan.  This happens on a 1 on a D6 and causes the Mongols to withdraw.

When a Mongol general element is lost in a battle or siege or the khan dies the Mongols withdraw and the Koreans roll for all captured cities - 4,5,6 they return to Korean control before the Mongols return.

After the Koreans submit or the Mongols withdraw (due to the death of the khan, the Koreans suing for peace, or the loss of the Mongol general in a siege or battle) there will sooner or later be another invasion.  A new khan will want to finish the job; the Koreans will be caught in non-compliance with the terms; or an new prince will be picked out to command after some infighting.  Roll D10 for how many years pass before the next invasion.  The Mongols start with all the areas they have captured and their army starts in the north again.  The Korean army may start anywhere and so may be able to retake some Mongol-held areas before the Mongol army arrives.

DBA battles are fought between however many elements are present - initially 12 but reduced by sieges and previous battles this year.  All are returned to 12 elements before Spring.  The exception is that Mongols may substitute in one element of Korean 7Hd per area held for any Mongol element to represent collaborators up to a maximum of four elements.

For reference, the Great Khans are listed below and events can be woven around them.
Ögedei Khan (1229–1241). 
Töregene Khatun (as Regent while Shiremun, Temüge  & Guyuk dispute the throne) (1241–1246)
Güyük Khan (1246–1248)
Oghul Qaimish (as Regent while Shiremun & Möngke dispute the throne) (1248–1251)
Möngke Khan (1251–1259)
Ariq Böke (1259–1264)
Kublai Khan (1260–1294)