The car, the key; ::: +++ SPOILER +++

When he felt bad, John Lynge left the car in order to go to a hospital; when he came back he realized that the car had been stolen. He thought that he had lost the keys. - In this episode Lund rejects the idea that the murder might have found the key by coincidence. The murder was planned since long, well prepared, etc., and this does not fit to such a coincidence. Okay.
Later on, Troels Hartmann drove the car to the party's flat. There he decided to put an end to his miserable life, and got to his weekend house by taxi. He left the keys of the car on a table in the flat.
There Vagn found the keys. It was by coincidence, wasn't it? How does this now fit to this crime?
And, even more important: How did Vagn know that these keys were the keys of that car?

English is not my first language. Any corrections are welcome.