I just got the Immortal badge. Really depends on your build.
I went afk on an ITF +4/+8. Got it down to roughly 400-500 dmg/sec.
Had a pocket emp with target heal on auto and radiation therapy on auto.
Used 5 Bosses, 4 lieuts, and a surgeon.
Still took approximately 50hrs for about 70mil of dmg.
I just did it every night when I went to bed for about a week.
For the Jailed badge, start an RWZ mission and get about 4-5 chief mentalists, the lieut versions. You will be held.