Well, you see how it works I assume. You will need to describe what "fail to engage means"? We don't know if the starter spins and motor does not or starter does not spin and so on. Be very specific.
There is a mechanical component and an electrical component. The mechanical is simple to check. Does the "plunger switch" get depressed or not. Is the linkage so loose it slips off the plunger head or does not fully press down? or hangs up somehow now and then?
The electrical has many more areas that could fail.
1. The battery
2. The battery connections/cables at battery/at starter
3. The coil connections/wiring
4. The side terminal and it's connections/wiring
5. The plunger switch mounted on starter (large push button assy)
6. The ignition switch and it's connections/wiring
7. The starter and/or it's drive
8. The flywheel
9. The copper slug on starter that contacts the plunger switch
10. Any and all associated ground paths
This list will be narrowed by your detailed explanation and observations. "All the better to serve you with my dear" said the wolf.
BTW: Is "second time in two weeks" the only two times you tried or 2 out of a 20? When it fails, does the next try work or many tries don't work? This is how you trouble shoot. Analyze and use deduction and know how the system works. Try not to over look anything. In general: Categorize electrical, mechanical, fuel, compression, and all that, to see what "system" it might be. We are glad to help but remember we can't see, hear, feel or test it from our house. We need you. You will figure it out. We are just holding the light.
