My girlfriend managed to fill 100% of her / partition on Ubuntu, and now it won't boot.

I can get into recovery mode and get a root console, and I've already used rm to delete several large files from her downloads directory, but the problem persists, even after several reboot attempts. ls no longer shows those files in the directory, and it should have freed up plenty of space to at least boot, but it's still not booting, and when I try to use the recovery mode option of 'fix broken packages', dpkg tells me it can't complete the action because it doesn't have the required ~250MB of free disk space. But, again, I just used rm to delete several hundred GB of files. That should be more than enough, but it's like the space isn't being freed up somehow. And, to be clear, I've done several hard reboots after deleting these files, still getting the same results. du -h does show her / partition at 100% usage, even after these deletions. (And the /boot partition is only at 49% full, so that shouldn't be an issue.)

Is there some command I need to use to get it to rescan and realize that it actually has free space now?

How much free disk space should Ubuntu actually need in order to boot, anyway?

Am I barking up the wrong tree entirely somehow? Booting always proceeds as normal right up until the moment the GUI desktop should load, and then it's replaced by a single blinking underscore in the top left corner -- it briefly shows any text typed in, not long enough to actually enter any command, but otherwise doesn't respond to anything I've tried other than Ctrl+Alt+Del, which reboots it. I've been pursuing this free disk space problem because the only unusual thing she did before this problem started is that she downloaded a lot of videos and filled the disk completely.

Edit:

No, the /home and / directories were on the same partition, so that wasn't the issue.

Okay, so I managed to get it to boot, I guess? For the possible reference of others, here's what I did:

1: Used rm to delete even more things from her download folder. Deleted a lot of stuff. The ls -lS | less command helped here, to quickly identify the biggest, most space-hogging files. I basically just deleted the 10 biggest files before trying again. (After already having deleted some of the other big files in there.)

2: I used exit to go back to the recovery mode menu, then chose the option to resume normal boot. That once again brought me to the dreaded blinking underscore and nothing else.

3: As some here suggested, I then used the Ctrl + Alt +F3 key combination ... and that just magically fixed everything. As soon as I hit that key combo, the screen flickered for a bit with weird graphical glitches straight out of a Cyberpunk hacking scene, and a moment later, it was suddenly fully booted into the normal desktop GUI.

At this point, I think I'm going to declare it 'fixed' and not fuck around with it any further -- I don't want to risk making things worse and losing this progress. Still not sure what actually happened, but it's working for now, and I'm a big believer in "If it ain't broke, don't fix it." Now that she has a GUI to work with, I'll ask her to move more of her downloaded files to an external drive, and I'll work on it further if and when something goes wrong again.

Thank you to everyone who offered suggestions and tried to help!