There are 2 well known strategies to farm Legendary Loot Midget (LLM). The first one can be way more efficient but require more luck. The second one offer always 3 LLM but require some work and a pretty long run every time.
MorningAfterKill made a quite good video about how to efficiently (not sure its a real word) use the 1st method :
When you receive the Mission Rocko's Modern Strife (By Brick), you can go to Thousand Cuts and No enemy will spawn except for special ones (including legendary loot Midgets). It's then really easy to open all the boxes you will find in the area. If you are lucky you cant get 1 LLM every 6/7 Min without being bothered by other enemies.
The other known solution is to farm them at wildlife exploitation preserve. The trick is to Never finish the "Doctor's order" quest so you can do it several time. You run all across the building to the hall where they experiment on Skags. In this place you have a room with 4 boxes, 3 of them always contain LLM if the mission is active.
(the guy talk a bit too much but you'll get the idea)