IIRC, instructions on the sleep mode are in the owners manual.
Once a battery has been drained down to a low voltage, it will never be the same...the more times you "kill" it the less it holds its charge, doesn't take a full charge, etc. Most battery manufactures consider 10.5 volts really dead.
Best battery we have found for our work vehicles is the Sears Platinum (made by Odyssey and re-labeled for Sears). Non pro-rated warranty, no acid leaks and very good quality. The Platinum won a battery shoot out in "Overland Journal" magazine, a very technical test that shows lab data and results (Optima didn't do well).
I'd start with a new fully charged Platinum and put the car in sleep mode. Also, absorbed glass mat batteries like the Platinum like to be charged at a higher voltage (around 14.7 IIRC) than wet lead acid batteries so you need to make sure your battery tender has the ability to charge AGM batteries. I've been using the C-Tek brand...so far, so good.
Cheers,
George