Forgot passphrase (encrypted hard drive), how can I use my computer again?

If you can't remember the password, then that's all she wrote.

All you can do is install a new OS. You don't need to enter recovery mode to do that. Just boot up the live USB and carry out the normal installation process. The hard drive will get overwritten.

I have Ubuntu on a USB, but when I enter recovery mode before I can reinstall Ubuntu, I obviously get asked for the password. Is there anyway around this?

I think you're missing a key step - boot from the Ubuntu USB drive. As it is, it sounds like you're not booting from the Ubuntu USB at all, and only getting stuck booting from the encrypted hard drive.

There should be a boot menu or a setting in the BIOS/setup menu to change the first boot device away from the internal hard drive and to the external USB.