To me it sounds during the upgrade when power was lost, important files critical to system function were corrupted due to being incompletely written. I'll provide some advice that I think might work. I don't know everything about Ubuntu so I apologize if some methods are impossible.
This method assumes you have 2 machines networked together that both support SSH and have SSH enabled.
First, you can try by accessing a TTY shell (Ctrl+Alt+FunctionKey), if it works you should appear in a text-only interface that prompts you to login. From there, you can input your username and then password, don't panic if you don't see the password being displayed on screen. If you can get in, you can use a TTY session to copy files you need via the
scp command. If you can't login and you have typed everything correctly, then there might be something wrong with PAM (Pluggable Authentication Module, used to verify passwords for instance) or another service.
This method assumes you have 2 computers and a USB drive.
If you have another computer to work with and you have a spare USB drive, you can burn an Ubuntu image onto it via a program like balenaEtcher. After doing so, plug it into your machine and start it up.
(NOTE: If you use an Nvidia GPU you might want to enter the GRUB config, enter the editing mode and attach
nomodeset onto the end of your /boot/ line then boot, otherwise there's a chance the machine will lock up during this process. It will look bad but it will work)
After doing so, you can enter a live session (do not choose to install), then you can open a terminal and run
lsblk to list all drives. After finding the drive with the data on it, you can mount it using the
mount command. You then can navigate onto the drive from Ubuntu's file manager, and upload files to a service like Google Drive or a NAS (Network-Attached Storage) device.