The problem with that solution is, the DVI-to-HDMI cable does work...if I'm at the boot screen or (probably) in Ubuntu mode. It's only in Windows mode where the CPU isn't acknowledging it. The DVI port is the only one that results in the boot screen getting displayed, which means that cable is the only way to get into Windows mode.
I regret to say I'm stumped. I think your plan to procure a monitor is probably the best bet. (Bff would like to add that secondhand monitors are cheap as fuck.)
Progress! Dad scooped Grandpa's old monitor, and I hooked it up via the VGA cable. (And unhooked the extra HDMI cable.) Result: two working monitors in Ubuntu mode! And one working monitor in Windows mode! This led to the discovery of what was going on in Windows mode where it was just hanging forever and then rebooting itself, though: it was trying to do a system repair and failing. I managed to progress past the system repair screen, and get to the advanced options screen; trying to skip past all of this and start Windows 10 anyways does...not work, the computer just restarts itself and then goes back into system repair when I try booting up Windows mode again. I'm currently having it try a system restore, which is gonna be kind of wild since the earliest point it gives me as an option is from October of this year, back when I first tried booting it up. So we'll see what happens. Worst case, I should have a functioning Ubuntu setup and I can learn how to update that and get it working so that I can play video games that require Windows?
Aaaand system restore failed! Error message is "System Restore failed to extract the file (D:\Windows\System32\drivers\wd\WdBoot.sys) from the restore point. An unspecified error occurred during System Restore." I'm assuming that this means that Windows has managed to eat the driver that tells it how to start up. I'm gonna need to reinstall Windows, aren't I?
Okay, current plan is less drastic than a complete reinstall of Windows. I'm gonna try and get a copy of the file onto a USB stick, and see if I can transfer it from the USB to the desktop. Ideally, this will fix the problem and not create new ones. Given the current track record, I'm not betting on that second part, but...we'll see? Gotta find a USB stick first, to try this.