According to John of Plano Carpini, if you changed horses five to seven times a day, the journey from Sarai to Karakorum took around two months. Between Kiev and Sarai John only changed horses three or four times a day, and it took about three weeks to cover that distance. With a large army, it would probably take longer. John estimated that for an army to reach "Cumania" (Modern Ukraine) from Karakorum would take three to four years, but remember that he never saw a Mongol army on the move, and in my estimation that timeframe is more likely between four and six months.