Try putting it in DFU mode. Hold down the home button and then plug it into the USB cord attached to your computer with iTunes up. That should kick it into DFU mode and iTunes will detect that it needs to be reset and will go through the steps of restoring it again. Hopefully that'll fix it.
The white screen, also known as white screen of death, can be caused by a couple of things after a fall. If you're lucky all that happened was that the LCD connector came disconnected and you just need to lift the screen up and securely connect it again. If you're not so lucky it means that your LCD is now not working. ifixit has a step by step for replacing the LCD. Try following steps 1-4 except when you get to step 4 try pushing that connector back where it belongs instead of unattaching it. Front Panel Assembly Instructions Good luck! Let me know what happens.
I did this replacement also and on the power button at least I found out that the metal loop attached to the power button has to be down for it to work correctly. I put it up and it only stuck out about half as far.
Was it working before you jailbroke it? Has it had some kind of catastrophic physical event happened to it (ie dropped)? If it was working before you jailbroke it and it stopped working afterwards I would say that it's a software problem and you should try restoring it (unjailbreaking it).
All of those things are connected together off of one ribbon cable. The cable may have come unplugged when it fell. To get the iPhone open turn it off, stick a suction cup on the touch screen right above the home button, and pull on that suction cup straight upward. This may require a little bit of force. The touch screen and the LCD should all come up together. There are three ribbon cables connecting the screen assembly to the rest of the phone. Read through Removing Display Assembly to see pictures and instructions on how to do this. Once you've gotten that off you can look at the #5 connector. It should be labeled with a sticker saying "5" with an arrow pointing to a connector. Check to see if that connector has come loose. If it has push it back onto the pins that it sits on and then close the phone back up. Hope this helps. Post a comment to let me know if this fixes it. iPhone 3GS Headphone Jack Replacement
Which iPhone model do you have. If it is is a used one there is a chance that it has gotten a lot of pocket lint/dust stuck underneath the home button (happened to a lot of them). You can try getting a can of that canned air used for cleaning out keyboards and such and stick the nozzle really close to the outside of the home button and spray in there for a little bit. It may move the dust away from the contact and you might be in business for a while again.
It really isn't necessarily a hardware problem. Try getting the phone into DFU mode (hold down home button and then plug into USB plugged into computer) and restore it. If that doesn't work there's another thing you can try but it's kind of complicated. Let me know if it works.
Connector 1 which is a pain in the you know where to get seated is more than likely what is giving you the problem here. It's hard to tell if it is completely seated or not. I would suggest pulling the screen off again and checking connector one to make sure it's fully seated in there. You might also want to try letting ifixit know that the LCD is not working they might send you a replacement to test out and see if the product is faulty.
You could try doing a reset by holding down the home button and then plugging it into the USB cable plugged into your computer. That might turn it back on. You will have to restore your iphone when you do that though so if you don't have it backed up don't try that.