AlwaysAisle

I think that will depends on how much luggage you have. There is an airport bus departing directly from Sheraton Miyako Hotel Osaka. There is a bus departing 8:10 a.m. and arriving at KIX 9:01 a.m. which should be plenty to catch your flight departing 11:00 a.m. from KIX.

If you want to take a train then choice is:

Subway Sennichimae Line from Tanimachi 9-Chome station to Namba station

Or

Kintetsu Line from Osaka-Uehommachi station to Osaka-Namba

Subway Sennichimae Line from Tanimachi 9-Chome station and Kintetsu Line from Osaka-Uehommachi station are pretty much same located at the ground level of Sheraton Miyako Hotel Osaka.

Then from Namba take Nankai Line to Kansai Airport.

There may be little bit of walk transfer at Namba station. I say amount and size of your luggage will determine bus or train.

Fare for bus is 1550 yen.

Fare for train is (assuming taking express train Rapit) 1610 yen if you take subway to Namba, and 1580 yen if you take Kintetsu Line to Namba.