LOL, when I tried a number of diffrent styles and timed the results I found the fastest way for me to solve L6E was to just compleate F2L and then do ELL
It makes a pure 2-look, you can erasily see how to solve centres and BD+FD in one look. Recog for ELL can be slow and some algs are terrible but it is still fast becuse you only stop once.
Another method:
1; pair up UR and UL in D and then palce them in correct position.
2; EML, edges of middle layer, usally in 2 looks.
2a; simply use U2 M moves to solve one of the oriented edges if any. 4-flip is bad but that only happens 1:8 times. If it does you have to use a 3-cycle or a pure flip also for the first look (or learn all 4 algs you need for one look if it is 4-flip, I'm currently looking for them =))
2b; solve what's left, either a 3-cycle or a 2-flip.
In 1:8 you skip EO and have the usual Roux end after step 1.
Really smooth actually =)
3-cycles:
U2 M' U2 M / M' U2 M U2
U M' U M' U2 M U M (or mirror)
(y) U M U' R2 U M' U' R2
(y) R2 U M U' R2 U M' U'
Edit : 4-flip (not pure) is 11 STM optimal:
(U F2 U2 M' U) M2 (U F2 U2 M' U) (11f*) (E perm)
More edit: got a pure in the same length:
(F U2 F2 M F) M2 (F U2 F2 M F) (11f*) (pure)
(F U2 F2 M F') M2 (F' U2 F2 M F) (11f*) (x perm)
The last case (3-cycle 1-flip) is bad, it takes 16 MU optimally. Use a 2-flip on the permuted edge and one more and you have a 3-cycle.
Optimal face: R F' D' F2 B' R' U F U' F2 B D R' F (14f*) ... hmm,

Optimal MU: M' U' M U' M' U' M U' M' U M' U M U' M U' (16,16) ... q-moves!
Optimal R2/Rw2 MU: M' U' (R2 M') U' M' U Rw2 U R2 U' M' U Rw2 (14s, 13a) ... better.
Optimal F2 MU: F2 U M2 U M' U2 F2 M2 U' F2 M F2 U (13,20) ... works but Rw is faster I think.