the 85 ... blows the doors off the 70-200 for image quality
I disagree... I think the 70-200mm f/2.8L IS II wins in the 'general' IQ department - similar sharpness in the center and sharper away from the center even with both at f/2.8 (the zoom wide open, the prime stopped down which should give it the advantage, but it's not enough), less vignetting, similar lateral CA and less axial CA. The 85L certainly doesn't 'blow the doors off' the 70-200 II, IMO. But this discussion isn't really about 'general' IQ.
I don't need any advice like "my lens is sharper" or "the 85mm has a lot more/pleasing bokeh"
But...the 85L II
does deliver more pleasing bokeh!

I also own both, and have used both on APS-C and FF - and my recommendation is different depending on sensor format. On APS-C, the 85mm focal length is great for tight portraits indoors and works great outdoors as well. The f/1.2 delivers excellent subject isolation. Honestly, on my 7D I found the 70-200mm focal length to be a bit awkward - too long indoors, and the f/2.8 on APS-C often didn't deliver sufficient background blur for subject-popping portraits (and also, not long enough when I needed reach outdoors - but, I have the 100-400 for that). For your 7D, I definitely recommend the 85L II. For outdoor use, get yourself a 72mm 3-stop ND filter so you can shoot wide open.
On FF, I think it's a different story. An f/1.2 on APS-C is like f/2 on FF - f/1.2 on FF often gives too shallow a DoF, so the 85L is often in the f/1.6-f/2 range anyway. The 85L still delivers magical shots, but the 70-200mm range on FF is a lot more versatile indoors and out, and f/2.8 delivers a nice, shallow DoF. If you were shooting FF, I'd likely recommend the 70-200 II
unless your main (only?) use for the lens was portraits...in that case, it's hard to beat the 85L II.
Here are a couple of examples of the 85L II on the 7D:

EOS 7D, EF 85mm f/1.2
L II USM, 1/2000 s, f/1.6, ISO 100

EOS 7D, EF 85mm f/1.2
L II USM, 1/1250 s, f/1.4, ISO 100