OK getting a wireless connection up in recovery mode is a PITA, so let's do this from a chroot environment.
Boot to a Peppermint LiveCD/LiveUSB .. set up your wireless connection, so you have internet access.
Open a terminal and run:
sudo mount /dev/sdb5 /mnt
for i in /dev /dev/pts /proc /sys /run; do sudo mount -B $i /mnt$i; done
apt-get install --reinstall xserver-xorg-video-intel
Now hit Ctrl+D
(to exit the chroot environment)
Reboot and check if Peppermint (on the HDD) will boot normally.