yes, this is difficult to manage. my advice would be:

1) take the quick release fully out of the kickr and place it on the ground right beside the unit
2) with your rear wheel off, hold the bike's saddle with your left hand and kind of get the front wheel where it needs to be
3) with your right hand, pull the chain backwards away from the rear dropouts as you lower the rear end towards the kickr
4) wrap this chain you've pulled around the smallest cog on the kickr cassette
5) slide the rear dropouts onto the kickr

it's actually pretty easy once you get used to it. you'll always get your fingers a little dirty though. i can make a video of this, but this is also how i swap wheels when my bike is on the ground and not in a repair stand.

Thanks for the reply. Everything I have seen advises putting the bike in the small ring up front and smallest gear in the back.

Do you find is easier to leave the bike in the small ring up from and large ring in the back? If so I'll give it a try.

whoops! typo. corrected. put the chain on the smallest cog on the cassette! you get a little more slack when the wheel is in the smaller chainring, but i've been able to do it just fine in either.

My TT rig has horizontal dropouts and I never remove the QR to put it on or take it off the Kickr. Gear to small ring/smallest cog.

It takes a little time to figure out hand placement that works best, but I always put one hand on the saddle and with the other grab the chain and think it feels like pulling back on an archery bow when you're about to launch an arrow. I keep my drivetrain pretty immaculate, but you may want to keep a rag handy.

I do miss the simple days with a vertical dropout bike when you don't even have to touch the chain for this stuff