Dell Inspiron 1564 overheating but fan not switching on, how to diagnose

I've got a Dell Inspiron 1564 laptop that is about one and a half years old. Since about a week, the laptop started to overheat, causing it to switch off unexpectedly…

The cpu fan is working erratically, it can start to spin for a while, doing its job and cooling down the cpu before it stops, but then the temperature goes up, and the fan doesn't reacts, once the temperature reaches a critical point (over 85 celsius, checked with speedfan…), the laptop switches off…

I already cleaned the vents and fan from dust, to no avail, and it was actually quite clean anyway.

Drivers and bios are up-to-date, no crapware was ever installed on this machine.

I don't know how to diagnose the problem, could it be the temperature sensors that sends wrong information, so the fan doesn't reacts? but then I believe the computer wouldn't detect the overheat and stop…

Is there a way I can pin point the problem? Maybe some low-level diagnostic tools to check functionality of sensors and fans???

The warranty is already over so any suggestion would be welcome.

Thanks!!