Does it turn on? Does the green light stays while there's no image on the screen?
If the lcd is damaged, it could be a reason why it doesn't show anything, though could be a blown fuse but since you've changed the MB,i don't think it's the problem. The no sound part might indicate a power issue, the speakers are dead (less likely) or it needs a functional display in order to work?
You could try with another lcd replacement (be careful with those ribbon cables), if the system still can't power on after that, then get a new power switch board.
Honestly i would suggest getting another one, unless somehow it's not an option for you. You still have to find a screen replacement (and possibly other components), so getting a new working PSP is the option imo.