Closest I could find to a player of KD's caliber joining a playoff team right before the playoffs is Michael Jordan's return (from baseball) to the Bulls in 1995 - just as the playoffs were set to take place
"At first, Jordan wanted to return in the postseason. Jackson countered with 20 games; Jordan then envisioned 10 to 15. It would eventually be 17."
The Bulls eventually entered the playoffs as the 5th Seed (Suns are obviously 4th seed this year).
From what I gather, the Bulls beat the Hornets in the first round and then met the 1st seed Orlando Magic in round 2
Bulls lost the Orlando series in 6 games, even with Jordan back in the team, and even though Jordan was familiar with much of the roster
Special Note:
Phil Jackson was not fired after the second round loss