BEFORE YOU READ THIS KNOW THAT I AM JUST AN ELECTRICAL APPRENTICE I AM NOT A LICENCED ELECTRICIAN NOR CAN I GIVE YOU THE ADVICE OF AN ELECTRCICIAN, I RECOMEND THAT AFTER YOU THIS YOU AT LEAST ASK A LICENCED ELECTRICIAN WHAT HE OR SHE THINKS BEFORE YOU DO SOMETHING ELSE
Now I did not mean to come across as rude with the caps but electricity is serious business and I dont want someone to take my opinion for gospel and get hurt or killed, that being said I have a few thoughts on your dilemma.
GFCI breakers/panels can and will be finiky. Most GFCI equipment (plugs, breakers, ect) have to "see" a load on them before they will function. So basically if I understand what you have written correctly, you have tried to turn power on, just to see if it works and you dont see power with out a load. This does not mean you did anything wrong. You could try plugging in the kettle and putting power to it, if this is so it will function if not the GFCI should trip.
"The sub panel has a ground and neutral bus. Should I connect my white wire to the neutral bus and the ground to the ground bus in both panels which are also tied together in the main panel?" Yes you should this is in theroy how this should be set up.
"The sub panel has a ground and neutral bus" These should be separate in the spa panel (as opposed to connected like in the main panel) and, What I am wondering is that your spa panel may need a separate and independant ground in order to function correctly ( GFCI plugs GFCI Panels and Breakers have internal electronics that "tell" them how to function, with out "seeing" the proper grounding your GFCI panel may not be working correctly)Now Im sure you have reviewed the literature that came with your panel, look it over again, see if it specifies the need of an independant ground. Running one is fairly simple, you run a wire (no6 or larger from the ground lug on the sub panel outside to a ground rod (an 8 ft copper coated zinc rod with a point at one end availible at home depot or lowes) driven MOST of the way into the ground(except the last 4 in or so) and connect the two with what is called an "acorn" ( its like a clamp) if you do this make sure you have bare copper and a good connection at the panel and at the ground rod.
I again aplogize for the caps I just wanted to make sure I made clear that the best solution is to ask a licenced electrician, but I also wanted to be as much help as I could considering I do have some limited experience with things like this.
kirscp is correct, no 10 is only rated for 40 amps, check your brew kettle for its ratings, if it specifies 50 amps you need to kick up to no 8, however a 3 wire no 8 will work as you will have 2 hot legs (red and black) a neutral (white) and a ground (bare) the ground is not counted in romex cable as it is not considered a current carrying conductor