How would you advise your classes to handle this side suit and why?
The answer is rather complicated.
IF THERE IS NO OPPOSING BIDDING, this combination becomes a standard “double finesse” because of the 9 spot. The percentage play is to finesse for first one honor and then the other. Finessing for King and Jack is not really different from finessing for King, then Queen in this scenario. Your chances for losing only one trick are about 75%. [25% of the time, LHO will have both King and Jack; 25% of the time RHO will have both King and Jack—and that is when you lose two tricks. 25% of the time LHO will have Jack and RHP the King; 25% of the time, LHO will have King and RHO the Jack. So with the double finesse, you lose only one trick about 75% of the time.]
This answer changes if you increase the number of cards you hold in the suit. If you hold nine or more cards in the suit, your best shot is to cash the Ace and then lead low toward the Queen.
Sometimes you catch a singleton honor and your worries are over. Occasionally you catch a singleton King with RHO and you can finesse LHO out of the Jack on the next round. With nine or more cards, eliminating two of the opponents' cards by cashing the Ace first increases your chances of playing the best card on the second round. You only lose two tricks when all four cards or KJ3 are with RHO. The majority of the time, you will lose only one trick, occasionally none.
Some people will also cash the Ace first with 8 cards in the suit, figuring they have a better chance of guessing correctly on the next round.
If you are playing against opponents you KNOW will not duck from a King, your best shot is to play a low card toward the Queen on the first round of the suit. If LHO ducks and you know that means s/he canNOT have the King, then you try the ten. If it loses to the Jack, next time you run the Queen toward the A9 because you are sure that South has the King.
Obviously, if the bidding makes it more likely that one opponent has the King, you play accordingly. If the bidding strongly suggests that LHO should have the King, play the Ace and low toward the Queen. If the bidding strongly suggests that RHO should have the King, play the Queen toward the Ace. After the King covers and you kill it with the Ace, your 9 and 10 are equals against the Jack. Again, you only lose one trick.
For yet another complication, perhaps the bidding makes it likely that one or the other opponent has a singleton in this suit. If it could be a singleton King, cash the Ace. If LHO can have a singleton Jack, but not King, run the Queen toward the Ace and let it ride. Hopefully, you will smother the singleton Jack. Then you can lead the 10 toward the A9 in the West hand for two more tricks.