Long time AA but new to KLM and spent a lot of time researching upgrade to J topic. Only option I could find, without paying thousands of dollars, was day of departure upgrade to J. Here was my experience:
(1) AMS-EBB booked in Q class by a travel agent J upgrade cost $405 US. Arrived from ATL on DL and went to transfer desk (T6) kiosk agent said paid upgrades available and walked me to desk to wait for agent. Process took 10 minutes but required agent to place a phone call. When I went to the gate they still had 10 seats available and sold all for the same price. Fare class did not seem to be an issue. One guy was quoted $729 by an agent walking around gate with an iPad and got the lower price by the gate agent. All persons required the gate agent to make a phone call to process.
(2) EBB-AMS booked N class and cost was $392 US for upgrade. Check in agent was contractor, said there were 2 seats available and sent me to KLM supervisor three stations down to pay fee and get new boarding passes. Required multiple phone calls and 20 minutes.
(3) AMS-ATL on DL booked in T class. I went to same T6 transfer desk at AMS and cost was $400 US plus 25,000 miles. Was told cash plus miles was only option for DL upgrades to J. Worth every penny for a 10 hour flight.

OBSERVATIONS: Did not get an electronic offer at check in time. Checked Expert Flyer to get availability, got to the airport early, all flight upgrades were purchased 2-3 h hours before departure at the airport. Upgrades take time to process and the transfer desk or check-in desk was more suited to process. Despite other repsorts, I did not see anyone on board with an iPad that could process.