I've had this exact scenario happen to me. A FA spilled a glass of water on my laptop. I pulled the battery and tried to dry to computer off, but some of the keyboards no longer worked.

The damage claim was quite straight forward. Following the instructions of the FA, I contacted customer service with my details, the flight information and a description of the event. The computer was repaired by Sony and AA paid the bill (if I remember correctly, they reimbursed me). The whole process was reasonably hassle free.

As an aside, I had another trip a couple of days later and I needed a fully functional laptop (the repair would have taken longer). I ended up buying a new computer. AA covered the repair of the original laptop and did not provide any compensation for time without a computer.