Hi, my name is XXXXX XXXXX I'd be happy to help you with this.
Unfortunately I wish I had better news. Let me explain.
The "white screen" happens on these machines a lot. It appears to be a problem with the video or video card, but that's not neccesarily the case. The display shows the white screen on power up until the video is initialized. Normally bios should initialize it during boot. The fact that you see HDD activity IS most likely point to it being a video issue, but that activity if it only lasts for a short time could just be the drive spinning up and initializing itself. If you see it running for a much longer period of time, then most likely you're correct and the machine is booting which means it is the video.
With all that said, there is obviously a failure of some sort in the video or motherboard. You've already checked the ram, which could result in it not booting. You also mention checking for lose cables. Assuming you checked all the video terminations, then frankly there isn't a lot more you can do.
The only real option if you want to keep the existing machine is to replace the motherboard. Pretty much everything is built into it, so replacing that will replace both the video and motherboard. Your local computer store should be able to order a replacement, or I'm sure it would be easy to find online from several dealers.
I'm assuming it's not still under warranty? It may be worth calling Sony if you aren't sure.
If you don't care about keeping the same machine, in a lot of cases when there is a complete failure like this, it's often times more practical to purchase a new machine and use whatever parts can be salvaged out of the old. The all-in-one doesn't leave a lot of options for that, but the hard drive (and it's data) as well as the ram and maybe even processor may be able to be ported to a new machine. Sometimes you may find it's not even worth doing that, you might be better off just porting the drive and getting better/faster ram and processor.
If you have any other questions about purchasing a replacement just let me know.
Wish I had better news!