Oh boy, I'm looking at the shop manual for my 6L80... well, here's what it says:
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9 out of 10 transmission shops are going to want to either rebuild the entire transmission or replace it with a remanufactured one. A new GM reman and converter will run about $5-6K installed, maybe less depending on the market and dealer and comes with a 3 year 100K mile warranty and can be serviced nationwide.
If you want to know exactly what's wrong with it, get a bi-directional scan tool on it, like a Snap-On Solus or GMs MDI with GDS software on a laptop. Otherwise, you'll get the standard, "your transmission is bad, it's need to be replaced" answer.
As you can see, you might get away with a new valve body/TECHM and torque converter.