A known problem with the Sonata is something called the starter lever inside the starter. It is made of plastic and will snap with age. When broken it won't allow the starter to engage with the engine.
This lever broke on my 2011 Sonata GLS (conventional key start). Got it to the dealer who replaced the lever for under $250.
I've read reports on other forums about dealers replacing the entire starter for about $650.
Good luck.