Lately the laptop fan smelled burned and today it got black in the left corner and got hot. What is broken and how can I fix it?
Unscrew the laptop, clean the fan, replace the thermal compound, done. Helps in 99% of cases when the laptop gets too hot. If something turns black it sounds very unhealthy, something could have burned too. Then you have to be a specialist.
Steer clear of this, sounds like KabelbrAnd or similar. On, stop working and disconnect
Better to repair?
Are you doing gaming with the laptop? That would be the normal consequence, against which there's no tool. Laptops are not designed for high performance.
or a cooler / fan is totally dusty and then the cpu gets enormously hot. If you do not clean it up immediately, your lapi will soon endure the final heat death.