You've already been told what it is by now. If you need a replacement: ATHENA AT40 400W AT POWER SUPPLY Athena Computer Power AP-AT40 400W AT power supply: Computers for $51 US. It can be switched from 110V to 220V and is higher wattage than the one you have. Remember when plugging into the motherboard, the black conductors are toward the center.
I was about to suggest how to wire it up but getting one of these is easier: https://www.amazon.com/dp/B075ZPWJBX
Takes an ATX power supply's 20 pin connector, converts it to the two AT motherboard connectors, and the extra green/black go to the front panel push button.
I wrote an article about someone who cracked "fuzzy bit" copy protection. You might find it interesting: https://www.hackster.io/news/breaking-80s-dungeon-master-disk-copy-protection-with-a-vintage-bbc-micro-b9da5a27e079
Older ATX PSUs actually do put out -5V, I used one on the inside of a PSU for 5150. I wired the big red switch to the green/ground from the ATX PSU, then just cut and spliced on an AT header to the correct wires from the ATX header. Worked quite well.
But, that said you can buy a brand new 230W AT PSU made by Startech on Amazon. It has the -5V out. I'd suggest replacing the innards of your failing 5155 PSU with the innards of the Startech PSU.
I'f they aren't snapped off you can probably bend them back.
Try a sewing needle or prescision tweezers like these : https://www.amazon.com/Precision-Stainless-Electronic-Anti-static-Industrial/dp/B08DN6MYTP
I have had a lot of luck with cheap NGFF (2 slots) m.2 ssd's and 2.5 ide adapters; specifically this 12$ adapter:
And although I have not used this exact drive here is an ngff m.2 256gb for 18$:
I use this in PowerMac G4 MDDs, a G4 laptop and a Pentium 4 ide-only motherboard and they are all fast but act like a typical drive.
Good luck.
So from a quick Google it looks like that might be an ATI VGA graphics card - does to connector have 15 pins - like this? What are you connecting it to for a monitor?
As far as whether the computer is running, I'd try a "isa post tester / diagnostics adapter" - here is one example on Amazon - they are only like $10 from most places and super useful for troubleshooting.
I was actually pondering replacing that battery since I have some parts coming my way.
As for the drive adapter… that was the easy part. I did have to write an image to the SSD using Etcher but that also allowed me to pre-install everything I wanted using a faster (and honestly more capable) emulator. I used one of SavageTaylor’s blank 2GB HFS images as the base. This is the adapter I am using
Go to amazon and search for 2gb usb drive. There are many bundles of 2gb drives for arouns $20-40.
20 pack for $45.99 and smaller sizes available.
https://www.amazon.com/ALMEMO-Drives-Swivel-Memory-JumpDrive/dp/B07SCR8G3M/ref=sr\_1\_9?crid=1YV3R0OBEWQ3Q&keywords=2gb+usb+drive&qid=1653864605&sprefix=2gb+usb+driv%2Caps%2C115&sr=8-9
Like others have said, just grab an appropriately-sized heat sink from somewhere (buy one, grab an old CPU heat sink, pull one off a dead card… whatever) and then I’d recommend some 3M thermal tape to hold it on. You can get a roll of it on Amazon for under $10.
If you're like me, and you don't have a c64 emulator, and you want to view this in-emulator:
ROM: http://csdb.dk/release/?id=159876
Emulator: http://vice-emu.sourceforge.net/index.html#download
http://www.oldversion.com/ has worked for me in the past for a lot of stuff like that. However it does tend to be very English dominant. Good luck!
Microsoft is usually pretty good about keeping old downloads alive but you might have to go searching for them as they’ve redesigned their website a few times.
I second everything that others are saying here. I would also suggest <http://www.linuxfromscratch.org/>. The non-systemd version would be the most useful for your purposes.
EDIT: Is there a BSD analogue to LFS?
I think it should be doable, though I've certain never tried.
FWIW you may have a hard time finding suitable hardware that support PS/2 keyboards+mice so if you can go with USB do so.
https://www.startech.com/Server-Management/KVM-Switches/8-port-Enhanced-Digital-KVM-Switch-over-IP~SV841HDIE
^ this sort of thing might work, but is VERY expensive.
If you can find ethernet based extenders for the keyboard, screen, and mouse you might be able to combine that with a more conventional KVM. Alternatively look at KVM over IP solutions that have nothing but ethernet ports and require a module on the other end with plugs on one side and an ethernet jack on the other might be good.
A software solution might be better. If it were modern computers then this would be good possibly, but it only supports Windows 7 and later. For Windows only, back to 9x you could try using UltraVNC.
I think the retro computers bits throws in more of a snag than you'd expect.
Solderings not particularly hard. Just use decent quality stuff. Using cheap crap is a surefire way to wreck your stuff. You want a decent iron preferably adjustable. The reddish orange weller adjustable one you can get for 40 bucks at most hardware stores will work well enough between dial setting 4 and 5 works well for boards. A good flux. The most important part. A good flux can make bad solderwork decent. And good leaded solder. Dont try to use the lead free stuff its much harder and doesnt offer as good a result. Practice on some scrap electronics you were going to throw out. Do not use plumbing solder. It has acid in it to clean the pipe and will eat through the wires on the board. This looks like a functional replacement for the battery in that. Just make sure the voltages match up and that you get the polarity right. If you hook it up backwards at best you will confuse the hardware and at worst kill it. https://www.amazon.com/Rechargeable-Ni-MH-3-6V-Battery-Varta/dp/B08FRBS44M. If for some reason you dont want that one search for 3 cell nimh cmos battery and you should get other results. Just check the reviews if possible so you dont get sent crap thats been on a shelf for 20 years.
I'm not familiar with this model, but how handy are you? This guy was able to build his own EGA to VGA converter without much trouble:
http://www.vcfed.org/forum/showthread.php?59244-Homemade-EGA-to-VGA-conversion-circuit
Edit: Or maybe something like this? https://www.amazon.com/Zerone-Converter-Adapter-Monitor-Digital/dp/B07NNTXC9Y