I don't have a DX6i to test out but I managed to download a copy of the manual from Horizon. OK, what you need to do firstly is to create a new aircraft (plane or heli) for flight simulator use. If you are not familiar with how to do this, please refer to your user manual for the DX6i. The next step is to select "Heli" or "Acro" as the Model Type. Now, name your Model something that will be easy for you to recognize later like "Aerofly". Then do the setup for the aircraft.
I checked the manual and for the DX6i, there is no software setting to able/disable trainer mode. So, just leave it as it is. With the main power switch off, does the radio come on when you plug the adapter cord into the trainer port behind the radio?
If you managed to reach the calibration setup, there should be a connection established. There could be some intermittent connection somewhere along the cables. Did you push all the male/female connectors fully in? How does the mono plug fit into the trainer port of the radio? Does it fit snugly or there is still some gap between the plug and socket? Sometimes due to mechanical tolerances of the socket and plug design, there could be some intermittent connection. If you "jiggle" and push the plug at the trainer port, do you see Aerofly showing the connection going on and off?
The calibration screen will only show a default picture of a controller and not an image of the controller you're using. That does not matter. I suggest you read through the setup and calibration procedure in the Aerofly manual. I'm sure the DX6i works with Aerofly as can be seen here http://www.rcuniverse.com/forum/m_7886489/tm.htm
Please ensure that you have also updated to the latest version 126.96.36.199 http://www.aerofly.com/downloads_en.html