What kwblue said… LOL…
blackjackinfo is right about the power supply fan, that could fail too – but it’s a lot bigger than the CPU fan, and generally doesn’t clog – either it works or it doesn’t. And cjperotti is also right about the gel.
Strange about the processor apparently not in the right place – most sockets are built such that they cannot be closed if the CPU is not firmly seated. Which means the machine should never have worked at all… not knowing what type of processor/socket you have I can’t say for sure – but if the CPU and fan are not in a single housing, the more likely explanation is that the fan was not seated properly on top/aside the processor. That’s where the silicone gel comes in – so you may want to take off the fan and look to see that the silicone is not hard, otherwise the problem will eventually come back.
Why do I know? I used to build computers for a living… and specialized in overclocking, so I had to buy special fans and silicone gel, and even hunt for processors from a certain production lot!