Axel's mother, Linda Peanberg King, told St Pancras Coroner's Court that she and her husband Alistair made contact with the Harmoni centre in the Whittington Hospital in Archway, after their son fell ill on the night of October 30 last year.
Five days later - after consulting at least three doctors - he died from bronchopneumonia.
More top news
Arsenal managed to beat Everton at the weekend but it wasn't enough to finish in the top for the first time in Wenger's time at the club.
The entertainer was held for the murder of Stuart Lubbock, who was found dead in his swimming pool in 2001, but never charged.
Although the diamond was believed to be cut in the 19th Century, its history and how it arrived at the car boot sale is unknown.