Try turning the PowerBook on with only the built in RAM module in it (assuming that you have OS X Tiger 10.4 installed as Leopard has a minimum of 512mb). My girlfriends MacBook has problems turning on and it was due to a RAM issue.If the problem is still there try turning the PowerBook without the battery in (i.e run it off the power adapter).
The OS software may also be the problem so, if you have the correct disk(s) (either the black Tiger disk or a grey one that says PowerBook on it), slide it in the drive while the PowerBook is on and then turn it off. Then turn the PowerBook back on whilst holding the 'C' key on the internal or an external keyboard (just incase the keyboard broke when it fell). Hold the 'C' key until it boots to the installer of the OS - start holding it from before the chime sounds.
I had a similar problem with a PowerBook. It froze all the time and a lot of the time didn't turn on. I took it to an Apple store and they told me it was probably the hard drive but I managed to fix it by simply re-installing the OS.
If none of these work, buy a cheap SATA hard drive (between 20 and 100gb) and install the OS using the internel dvd drive.
Another more complicated way of forcing the laptop to have its operating sytem reinstalled is by connecting two Macs together (with the PowerBook in question off) using a FireWire cable. Then start the PowerBook whilst holding the 'T' key. It should come up with a yellow FireWire logo on a blue background. You then need to insert the install disk into the Mac that is already on (not the PowerBook) and run in the installer from that computer. however when you come to the part when you need to chose the hard drive, select the one from the PowerBook. I would advise you open the disk utility from the menu bar on the installer screen (I think it's under utilities) and erase the hard drive of the PowerBook. Then proceed to install the OS - make sure you select the PowerBook hard drive to install to.
As other people have said it may be a logic board fault. If that's the case then I would advise looking for a replacement PowerBook. In some way or another you should be able to get all your files off the PowerBook.
It also may be the case that it's none of these. Good luck! Please post your findings on here.