Well I fear we are almost as far in as we can go with this now, I don't think you have missed anything above, but the next bit is something I stumble around with more than know what\how to do, at least in the way to guide someone, but I will do my best, but here are 2 last desperate attempts, if these fail I fear you will have to take it for repair, probably a new motherboard, because the fact this is failing it appears to support the thought of a faulty board, and this is often the most impossible thing to be certain about, so sometimes it takes a leap of faith, and I fear we are about at that point now.
Here goes anyway, tried looking through the manual(s) and saw no bios settings anywhere, so what I need you to do is concentrate on the display as you power on, keep watching and reading the info that is displayed, you may need to boot several times to read it all, because it can flash by quickly, what I need you to watch out for is ANY mention of pressing certain keys to enter the bios set up utility, not that wont be the exact term, and is not to be confused with entering the bios settings, you are looking for a utility to aid flashing the bios.
Should that show nothing, then enter the bios settings as you have previously, now you are looking for the same thing again, I don't know the exact name, but you again are looking for anything to do with a utility for updating the bios, if you find this via any method run that then direct it to the optical drive for the files it needs.
Now if nothing shows there I can only think of one last desperate attempt to do this, and this is where I start to stumble around, the first thing we need to do is enter the dos part of the system, this appears like the bios setting (black background etc), and requires the use of dos commands to move around and do things.
So first thing we need is to get a boot disk, again this is an area I am not that clued up about, so hopefully from what little I can advise and what you can achieve hopefully we can get you there.
While I was looking for a boot disk, I was unable to clearly identify the best disk for you, but, I did come across another file that showed some promise, so heres hoping it can help.
Go Here, look for the "BOOTABLE CD" section, it's in the top part of the list in the second paragraph kind of thing, under that heading there is a 2nd section called "Need To Flash Your BIOS On A PC With No 1.44 Read.1st" that is what you need to look for, hopefully if you use that following the read me section you will be able to make that disk, I just wish I could guide better on this but it's not something I have got experience with, if your lucky you can create a disk with the files and flash via that.
Alternately (and what I was going for before I saw that disk) you need to get the system into dos mode, eg a black or blue screen with text only, and all it will show is C:\> or something along those lines, then simply using dos commands direct the system to run the flash file.
Again this is a bit over my range, but you first need to change the directory your in, "cd" is the command line to do this, so you will need to type "cd D:\" (ALL quotes should be typed without the quotes), this assumes your optical drive is the D drive, if not replace with the correct designation, you will know if this has worked because the C:\ will change to D:\, now you need to run the flash utility on the disk, so you need to type "D:\run swinflash"
Now I may not be exactly right here, as I say I stumble around with this area, because it's many years since I have used dos command lines, and have probably forgotten small details that may make the difference between working or not, if you type a command that fails it will tell you, if this happens we will need to figure out the exact line required, sorry I can't be clearer, but I hope with what little I have been able to say helps a bit anyway.