Well, firstly, if your joy stick fell off. then go simply buy a new one on eBay or amazon, if it is broken take it off and buy a new one. its Easy. yes, its not broken if it fell off, just put it back on
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Bring your PSP to a game store (like Gamestop) and ask them to fix it.
delete the data and puted in your psp
Okay, this thing is disturbing,maybe you could buy a joystick replacement to fix it. By the way,online shopping may have more choice.
Go to the repair shop that it can fix your psp. If you don't see one find someplace else
Charge it...