It's not entirely clear from your description what's going on. Are you using the hair dryer to blow cool air into the fan slot? Then it could be system overheating due to a dead fan, or just a lot of dust & crud inside.
There are a number of utilities that will show you fan speed(s), CPU and motherboard temps, and much more. (gkrellm and conky are two popular in the Linux world.) You could try running one of those to see if it's shutting down due to overheating.
Also, you should be able to look up the meaning of those two blinking LEDs. Generally particular lights blink to indicate particular problems.