I've owned both, and yes, the 85L II is a lens that focuses ponderously, to be charitable. The 85/1.8 is much faster in terms of AF.
While it seems surprising that a $400 lens focuses faster than a $2000 lens, lens design is all about tradeoffs. In the case of the 85L, Canon designed the lens with a front-focusing mechanism (the 50L has one, too) - that means when focusing, the front element extends outward from the lens. In the case of the 85L, that's a big, heavy element that needs to move to achieve focus, which is why it's slow. The 85L also has electronic manual focus - turning the MF ring doesn't physically move any elements, it just activates actuators that cause the AF motor to move them (power off = no focus).
Bottom line, I would not recommend using the 85L II for indoor sports on a 5DIII (I might consider it on a 1-seires body - they have higher voltage batteries that actually drive lens AF motors faster - the 85L focuses noticeably faster on my 1D X than other bodies, but it's still not what I'd call 'fast').
When considering the 85/1.8, you should also consider three other lenses - the 100/2 (close cousin to the 85/1.8, trading a little reach for 1/3 stop of aperture), the 135/2L (arguably one of the best lenses for indoor sports, along with the 200/2L IS), and the Sigma 85/1.4 (faster focusing that the 85L, but Sigma is known for sometimes less than stellar AF accuracy and poor lens QC, so if you buy Sigma, buy from somewhere with a good return policy).
One other issue with using the 85/1.2L II in low light - if you use it wide open, the DoF is really, really thin. Only-one-eye-in-focus thin, depending on subject distance.
Given the focal range you mention - 90-140mm - I'd strongly consider the 100mm f/2. But...if 135mm is not too long for you, I'd pick the 135L over the 100/2 - the bokeh is a little better, and the AF is a little faster, too.
Hope that helps...