So, as you suggested it looks like you can't "reach" 2/3 of that space. You need to delete unwanted partitions then expand your root partition, currently /dev/sda6. So I would delete /dev/sda{2,4,5} then add all that space to your root.
This is only safely done from a liveUSB - either the Ubuntu install USB, or download gparted.
here is a link to what you need to do. You are fortunate thaat the swap is out of the way and won't interfere.
External disks are cheap - make sure you get a backup first.