Downgrade the kernel.
The kernel that came with 14.04.1 was 3.13.0-32-generic. The kernel that worked was 3.13.0-24-generic. It is available in the repositories.
How to get it?
Boot from a LiveCD/flash drive and open a terminal.
Now, we need to load ourselves into the computer's new, non-functional OS:
sudo mount /dev/sda1 /mnt
sudo mount --bind /dev /mnt/dev
sudo mount --bind /proc /mnt/proc
sudo mount --bind /sys /mnt/sys
sudo chroot /mnt
After the final command all commands you type are now executed as though you had actually booted up the computer. This is handy, because it means we can use:
sudo apt-get update
sudo apt-get upgrade
To grab the latest packages and install them.
We can think downgrade the kernel with:
sudo apt-get install linux-image-3.13.0-24-generic
sudo apt-get install linux-image-extra-3.13.0-24-generic
We can then be sure that GRUB is updated using:
sudo update-grub
If initramfs complains about things, you may need to run:
sudo update-initramfs
After installing the kernel and updating grub, reboot.
Start-up will probably fail again by showing a blank screen or one with a single blinking cursor in the upper left. Ctrl+Alt+Delete to restart a second time.
Now, the GRUB menu will show. Head to Advanced Options for Ubuntu, and boot into the older kernel.