Like most parts of life somethings suit one person but not another, I am of no doubt that brain training games work for some people but I am also certain they are a complete waste of time to others. Quite frankly I think things like this are more about improving memory, concentration and alertness.
Actual mental health really is at the mercy of nature, I have worked in numerous types of employment from management to manual labour to skilled labour to self employed and a few others in between, but I have always enjoyed reading and researching about anything that has taken my fancy or intrigued me, I have also always enjoyed planning and making things and even doing crosswords and jigsaw puzzles.
I had a mental breakdown despite this and I have since found that I struggle to do any of these things, but I still persevere and although it takes me much longer and that I have to read things in small bite size pieces I will not give up. So, I think it really is an individual thing and as for the onset of mental problems after retirement I feel a lot of that is caused by the fact that many of those retiring have never had hobbies or other activities that they do, they find themselves in a void where they have time but nothing to do with that time and can begin to vegetate or even just give up on life.
Therefore, my advice would be to have a hobby or an activity that you enjoy to keep your mind active and hope that life is good to you.