I doubt a tray was included for any iPad with your original purchase. If your 1608 was one of the older models they only had the 30 pin connector, at that time there was not even an iPad with a lightning connector made and no iPad Air made. The 1608 lightning came out at a later date.
You can get a replacement kit that installs on your 1608 to convert it from the 30 pin to the Lightning connector. That would work on a regular iPad. The iPad air2 is thinner and will not be snug in the tray so you would need to giggle it into place and take it off prior to moving the board. I guess you could shim it in tight with some cloth and then lock it down with the clamp that came with the 1608, but that would be a pain every time you needed to remove the iPad to download music or run wireless. If you could find a older iPad with the needed memory and upgrade to the Lightning connector on the 1608 you would be in good shape.
The adaptor would work, but remember the iPad will still not slide all the way into the tray with the Adaptor in place and the little cheap adapters don't work well. The real Apple adaptor with the cable on it will work. This is the least costly and easiest solution.
Another option (which is what I do, with the same set up as you) is keep your old iPad docked on the board.
Use your new iPad air2 as a remote (2nd) controller and load all your stuff on it. Using the Apple Air Express router, you can send music to your board from the remote (your iPad Air) iPad. The apple router has a
1/8 inch output that you can run to an input on the 1608. So what happens is that your iPad Air with all your songs is now wirelessly sending music to the router, the router picks it up and sends the music to an input on the board. It also allows you to walk the room and change the music, adjust the volume or pause the songs.
It's really pretty simple.
